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    <id>https://micahrj.github.io/posts/vst3</id>
    <title type="html">Simplifying the build process for vst3-rs</title>
    <published>2025-12-26T12:13:00-07:00</published>
    <updated>2025-12-26T12:13:00-07:00</updated>
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            <p><a href="https://github.com/steinbergmedia/vst3sdk">VST 3</a> is an audio plugin interface which is developed by Steinberg and supported by a large number of host applications and plugins. The VST 3 API comprises a set of C++ header files which contain definitions for structs, constants, and abstract base classes (used in a similar way to <a href="https://en.wikipedia.org/wiki/Component_Object_Model">COM</a>). I maintain a set of <a href="https://github.com/coupler-rs/vst3-rs">Rust bindings</a> for VST 3 which are automatically generated from the original C++ headers using <a href="https://clang.llvm.org/doxygen/group__CINDEX.html">libclang</a>.</p><p>Earlier this month, I released version 0.3.0 of the <code>vst3</code> crate. In previous versions of the crate, bindings were generated at build time, and users were required to supply both libclang and the VST 3 SDK as third-party dependencies. As of this latest version, the generated bindings are now published directly as part of the crate's source, and the build-time generation step and third-party dependencies are no longer necessary. This change both simplifies the setup process for downstream users of the crate and significantly improves build times.</p><p>These improvements were made possible by a recent change to the licensing situation around the VST 3 SDK, but they also required solving some technical problems regarding the output of the binding generator. I'll talk about both of these aspects below.</p><!--excerpt-->
<h3>Licensing issues</h3><p>The first version of the <code>vst3</code> crate was released in August 2023. At that time, Steinberg's SDK was dual-licensed, and developers had the option of using it under either the <a href="https://www.gnu.org/licenses/gpl-3.0.html">GPLv3</a> or a proprietary license. Selecting the GPLv3 license meant that developers could make use of the SDK in open-source projects, but those projects in turn also had to be distributed under a GPLv3-compatible license. The proprietary license required signing an agreement with Steinberg and explicitly prohibited the distribution of derivative works of the SDK.</p><p>My goal with the <code>vst3</code> crate was to enable developers to use the VST 3 interfaces from Rust in all the same situations where they might use them from C++, including in closed-source hosts and plugins. Distributing bindings under GPLv3 would not have accomplished this, and the proprietary license did not allow for distributing bindings at all. Due to these constraints, the solution I ultimately settled on was for the <code>vst3</code> crate not to include any actual bindings in the first place; instead, the crate itself was purely a binding generator (released under MIT and Apache 2.0), and bindings would be generated at build time from a user-provided copy of the SDK (obtained under a license of their choice).</p><p>In October of this year, there was an unexpected development: Steinberg <a href="https://forums.steinberg.net/t/vst-3-8-0-sdk-released/1011988">released</a> version 3.8.0 of the VST 3 SDK under the MIT license. This was exciting, because it meant that there were no longer any legal obstacles to directly publishing Rust bindings for VST 3; however, there remained some technical obstacles which had to be overcome.</p><h3>Platform inconsistencies</h3><p>The <code>vst3</code> crate's binding generator uses libclang to extract information from the C++ headers in the VST 3 SDK. This saves an immense amount of work compared to writing a C++ parser from scratch. However, there is one significant limitation, which is that the information provided by libclang is specific to a particular target platform, and it may or may not be valid for other targets. Definitions can vary due to explicit preprocessor conditionals or arbitrary platform differences in toolchain behavior, and the only reliable way to obtain this information from libclang is to run it once for each target platform.</p><p>In the case of the VST 3 headers, there were handful of concrete differences in libclang's output on different platforms which were resulting in corresponding differences in the generated Rust bindings. This didn't pose a problem when generating bindings at build time, since the generator would always be run with the same target platform as the rest of the build, but now that I wanted to run the generation step ahead of time, those differences had to be addressed somehow. The brute-force option would have been to generate and commit a separate version of the bindings for every supported target, but I wanted to avoid that if possible. The approach I took instead was to add a CI check verifying that the generated bindings for each platform were character-identical and then address the failures one by one until it passed.</p><p>Comparing the generator's output on Windows, macOS, and Linux, I found the following differences:</p><ol><li>Definitions were being written out in different orders.</li><li>The <a href="https://en.cppreference.com/w/cpp/types/integer.html">fixed-width integer types</a> in <code>&lt;cstdint&gt;</code> (<code>int8_t</code>, <code>uint8_t</code>, etc.) were being mapped to different integer type aliases in Rust.</li><li>A small number of integer constants (specifically the <a href="https://github.com/coupler-rs/vst3_pluginterfaces/blob/31d6eeba6daaa3e2a8bfbe3e7a90ca0b7fbfbc1c/base/funknown.h#L161-L202">result codes</a>) had different values.</li><li>Enumeration types without a <a href="https://en.cppreference.com/w/cpp/language/enum.html">fixed underlying type</a> (i.e., the <code>int32_t</code> in <code>enum E : int32_t</code>) had different underlying types.</li></ol><p>The first two on the list were straightforward to eliminate, as they were ultimately arbitrary and didn't correspond to any real functional difference between platforms. Difference #1 was caused by the fact that libclang itself was traversing the AST in different orders, and it was easily addressed by simply sorting the definitions before outputting them. Difference #2 was caused by the fact that the binding generator was mapping C++'s fixed-width integers to the FFI integer types in Rust's <a href="https://doc.rust-lang.org/stable/std/ffi/index.html"><code>std::ffi</code></a> module, which was a totally unnecessary indirection — while the fixed-width integers map to different Rust FFI integer types on different platforms, and the FFI integer types themselves map to different Rust primitive types on different platforms, these differences always cancel out — and was addressed by mapping them directly to Rust's primitive integer types instead.</p><p>Difference #3 corresponds to a real difference between platforms, as the result code constants are actually defined to have different values on Windows than on macOS or Linux. I dealt with this by just providing manual per-platform definitions for these constants using <code>#[cfg]</code> attributes.</p><p>Finally, difference #4 was occurring because enum declarations without fixed underlying type default to using <code>int</code> on Windows and <code>unsigned int</code> elsewhere (<a href="https://github.com/rust-lang/rust-bindgen"><code>rust-bindgen</code></a>, which also uses libclang, suffers from the <a href="https://github.com/rust-lang/rust-bindgen/issues/1966">same issue</a>). This was the most annoying inconsistency to deal with. In principle, it might have been possible to deal with it entirely automatically, but libclang provides no reliable way to detect whether an enum declaration specifies a fixed underlying type or not. In the end, I dealt with this in the same way as difference #3, by providing manual type definitions for each of these enums.</p><p>Despite some annoyances, I'm glad I took this approach. It will be easy to keep the generated bindings up to date with new releases of the SDK, and the CI check should reliably catch any new platform inconstencies as they are introduced.</p><h3>Conclusion</h3><p>Version 0.3.0 of the <code>vst3</code> crate is <a href="https://crates.io/crates/vst3/0.3.0">available on crates.io</a>! If you're currently making use of <code>vst3</code> in any projects, I would highly recommend updating for the simplified build process and improved build times. Updating should be as simple as bumping the dependency version and then removing any setup for including the VST 3 SDK.</p><p>Additionally, if you're currently using the older <a href="https://github.com/RustAudio/vst3-sys"><code>vst3-sys</code></a> crate, I would strongly encourage looking into replacing it with <code>vst3</code>. <code>vst3</code> is not only more permissively licensed (<code>vst3-sys</code> is still released under GPLv3), it also contains a more complete and up-to-date set of bindings, and this will likely continue to be the case as its bindings can be regenerated automatically whereas <code>vst3-sys</code> must be updated by hand.</p><p>Finally, if you run into any issues, questions, or difficulties while working with the <code>vst3</code> crate, please feel free to file an issue on the <a href="https://github.com/coupler-rs/vst3-rs">GitHub repository</a>, post in the #vst3 channel in the <a href="https://discord.gg/8qW6q2k">Rust Audio Discord</a>, or start a topic in the #vst3 channel on the <a href="https://coupler.zulipchat.com/">Coupler Zulip</a>.</p>
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            <p><a href="https://github.com/steinbergmedia/vst3sdk">VST 3</a> is an audio plugin interface which is developed by Steinberg and supported by a large number of host applications and plugins. The VST 3 API comprises a set of C++ header files which contain definitions for structs, constants, and abstract base classes (used in a similar way to <a href="https://en.wikipedia.org/wiki/Component_Object_Model">COM</a>). I maintain a set of <a href="https://github.com/coupler-rs/vst3-rs">Rust bindings</a> for VST 3 which are automatically generated from the original C++ headers using <a href="https://clang.llvm.org/doxygen/group__CINDEX.html">libclang</a>.</p><p>Earlier this month, I released version 0.3.0 of the <code>vst3</code> crate. In previous versions of the crate, bindings were generated at build time, and users were required to supply both libclang and the VST 3 SDK as third-party dependencies. As of this latest version, the generated bindings are now published directly as part of the crate's source, and the build-time generation step and third-party dependencies are no longer necessary. This change both simplifies the setup process for downstream users of the crate and significantly improves build times.</p><p>These improvements were made possible by a recent change to the licensing situation around the VST 3 SDK, but they also required solving some technical problems regarding the output of the binding generator. I'll talk about both of these aspects below.</p>
        ]]>
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</entry>
<entry>
    <id>https://micahrj.github.io/posts/llq</id>
    <title type="html">LLQ: A wait-free SPSC linked-list queue with recyclable nodes</title>
    <published>2022-12-04T15:39:00-06:00</published>
    <updated>2022-12-04T15:39:00-06:00</updated>
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            <p>Last year, I published a Rust library called <a href="/posts/basedrop/">basedrop</a>, which implements a memory reclamation system tailored to the constraints of real-time audio scenarios. The purpose of basedrop is to make it easy to share dynamically allocated memory with a real-time audio thread while ensuring that no allocations or deallocations happen on that thread. This is accomplished by providing a set of smart pointers (analogous to <code>Box</code> and <code>Arc</code> from the Rust standard library) which do not directly free their associated allocation when dropped, but instead automatically push it onto a lock-free queue to be collected later on another thread.</p><p>Basedrop's design has some compelling benefits: it frees you from having to write code by hand every time you want to transfer an object to another thread to be freed, and if you restrict yourself to its vocabulary of smart pointers, it eliminates the possibility of accidentally dropping an allocation on the real-time thread (a mistake which can easily remain invisible if you don't have something like <a href="https://github.com/Windfisch/rust-assert-no-alloc"><code>assert_no_alloc</code></a> to catch it). However, after talking with some developers trying to make use of basedrop in real projects, it became clear to me that these benefits come at the cost of a somewhat opinionated API, making it difficult to integrate with certain program architectures. I decided that a stripped-down version of the core linked-list queue would probably have some value, and the end result of that was the <a href="https://github.com/micahrj/llq">llq</a> crate.</p><!--excerpt-->
<p>A central piece of basedrop's design is the <a href="https://docs.rs/basedrop/0.1.2/basedrop/struct.Node.html"><code>Node&lt;T&gt;</code></a> type, which represents a node that can potentially be added to the collector queue's linked list. Each of basedrop's smart pointers allocates a <code>Node&lt;T&gt;</code> on the heap at creation time, so when the time comes to mark the contained object as ready for deallocation, that node already exists, stored inline as part of the original allocation, and can simply be linked into the queue. This makes it possible to send an object back from the real-time thread to be reclaimed without performing any allocator operations.</p><p>The llq crate extracts just that core functionality, of a wait-free linked-list queue with preallocated nodes, and presents it in an unopinionated way. With llq, you can create some nodes:</p><pre style="background-color:#f6f6f6;">
<span style="color:#d35400;">use </span><span style="color:#395063;">llq::{Node, Queue};
</span><span style="color:#395063;">
</span><span style="color:#c0392b;">let</span><span style="color:#395063;"> x </span><span style="color:#d35400;">= </span><span style="color:#395063;">Node::new(</span><span style="color:#d35400;">0</span><span style="color:#395063;">);
</span><span style="color:#c0392b;">let</span><span style="color:#395063;"> y </span><span style="color:#d35400;">= </span><span style="color:#395063;">Node::new(</span><span style="color:#d35400;">1</span><span style="color:#395063;">);
</span><span style="color:#c0392b;">let</span><span style="color:#395063;"> z </span><span style="color:#d35400;">= </span><span style="color:#395063;">Node::new(</span><span style="color:#d35400;">2</span><span style="color:#395063;">);
</span></pre>
<p>push them onto a queue:</p><pre style="background-color:#f6f6f6;">
<span style="color:#c0392b;">let </span><span style="color:#395063;">(</span><span style="color:#cd4e38;">mut</span><span style="color:#395063;"> tx, </span><span style="color:#cd4e38;">mut</span><span style="color:#395063;"> rx) </span><span style="color:#d35400;">= </span><span style="color:#395063;">Queue::&lt;</span><span style="color:#c0392b;">usize</span><span style="color:#395063;">&gt;::new().</span><span style="color:#3092c6;">split</span><span style="color:#395063;">();
</span><span style="color:#395063;">
</span><span style="color:#395063;">tx.</span><span style="color:#3092c6;">push</span><span style="color:#395063;">(x);
</span><span style="color:#395063;">tx.</span><span style="color:#3092c6;">push</span><span style="color:#395063;">(y);
</span><span style="color:#395063;">tx.</span><span style="color:#3092c6;">push</span><span style="color:#395063;">(z);
</span></pre>
<p>pull them off the other end:</p><pre style="background-color:#f6f6f6;">
<span style="color:#c0392b;">let</span><span style="color:#395063;"> x </span><span style="color:#d35400;">=</span><span style="color:#395063;"> rx.</span><span style="color:#3092c6;">pop</span><span style="color:#395063;">().</span><span style="color:#3092c6;">unwrap</span><span style="color:#395063;">();
</span><span style="color:#c0392b;">let</span><span style="color:#395063;"> y </span><span style="color:#d35400;">=</span><span style="color:#395063;"> rx.</span><span style="color:#3092c6;">pop</span><span style="color:#395063;">().</span><span style="color:#3092c6;">unwrap</span><span style="color:#395063;">();
</span><span style="color:#c0392b;">let</span><span style="color:#395063;"> z </span><span style="color:#d35400;">=</span><span style="color:#395063;"> rx.</span><span style="color:#3092c6;">pop</span><span style="color:#395063;">().</span><span style="color:#3092c6;">unwrap</span><span style="color:#395063;">();
</span></pre>
<p>and even reuse them with a separate queue:</p><pre style="background-color:#f6f6f6;">
<span style="color:#c0392b;">let </span><span style="color:#395063;">(</span><span style="color:#cd4e38;">mut</span><span style="color:#395063;"> tx2, </span><span style="color:#cd4e38;">mut</span><span style="color:#395063;"> rx2) </span><span style="color:#d35400;">= </span><span style="color:#395063;">Queue::&lt;</span><span style="color:#c0392b;">usize</span><span style="color:#395063;">&gt;::new().</span><span style="color:#3092c6;">split</span><span style="color:#395063;">();
</span><span style="color:#395063;">
</span><span style="color:#395063;">tx2.</span><span style="color:#3092c6;">push</span><span style="color:#395063;">(x);
</span><span style="color:#395063;">tx2.</span><span style="color:#3092c6;">push</span><span style="color:#395063;">(y);
</span><span style="color:#395063;">tx2.</span><span style="color:#3092c6;">push</span><span style="color:#395063;">(z);
</span></pre>
<p>and none of the above <code>push</code> or <code>pop</code> operations will ever allocate or free memory, lock a mutex, or even enter an unbounded compare-exchange loop.</p><p>It's worth noting that essentially the only synchronization operations in the entire source of llq are a single <a href="https://github.com/micahrj/llq/blob/f5707bd832144308b3482c56b088b3076ea3dd25/src/lib.rs#L193">acquire load</a> in the body of <code>pop</code> and a <a href="https://github.com/micahrj/llq/blob/f5707bd832144308b3482c56b088b3076ea3dd25/src/lib.rs#L228">release store</a> in the body of <code>push</code>. I consider it a pretty compelling demonstration of Rust's type system and safety guarantees that a concurrent data structure with such minimal synchronization overhead can still have a <a href="https://doc.rust-lang.org/nomicon/races.html">data race</a>-free public API (assuming llq's implementation is bug-free, of course!).</p><p>For reference, the queue design in llq is based on a particular <a href="https://www.1024cores.net/home/lock-free-algorithms/queues/unbounded-spsc-queue">unbounded SPSC queue design</a> from 1024cores. There's also an implementation of a <a href="https://github.com/rust-lang/rust/blob/481971978fda83aa7cf1f1f3c80cfad822377cf2/library/std/src/sync/mpsc/spsc_queue.rs">similar queue design</a> in the internals of the Rust standard library.</p><p>It's important to note that llq is designed for a specific, uncommon set of requirements, and several aspects of its design are more or less the opposite of what one would want out of a general-purpose channel for communicating between threads. llq is designed for the scenario where</p><ul><li>you don't want the channel to interact with the OS scheduler at all (no blocking)</li><li>you want to ensure that one thread never allocates or deallocates in the process of sending or receiving items</li><li>you want sending to be infallible for one thread (for e.g. returning objects to be deallocated from the audio thread)</li></ul><p>A general-purpose queue should probably</p><ul><li>have blocking semantics and interact with the system scheduler, when receiving from an empty channel or sending to a full channel</li><li>store items inline together, for less pointer-chasing and less memory usage overall</li><li>have a bounded capacity, for backpressure</li></ul><p>So, if you're just looking for a generic SPSC queue, there's good chance llq is not what you want. But if you're implementing real-time audio software, or are otherwise facing a situation where some threads in your program have much stricter latency requirements than others, llq might be worth a look.</p><p>You can check it out over on <a href="https://github.com/micahrj/llq">GitHub</a> or on <a href="https://crates.io/crates/llq">crates.io</a>.</p><p>Discuss on: <a href="https://twitter.com/glowcoil/status/1599518887871008768">Twitter</a> · <a href="https://post.lurk.org/@glowcoil/109457572953430604">Mastodon</a> · <a href="https://www.reddit.com/r/rust/comments/zcm465/llq_a_waitfree_spsc_linkedlist_queue_with/?">r/rust</a> · <a href="https://www.reddit.com/r/programming/comments/zcmcrp/llq_a_waitfree_spsc_linkedlist_queue_with/">r/programming</a> · <a href="https://news.ycombinator.com/item?id=33858117">HN</a></p>
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            <p>Last year, I published a Rust library called <a href="/posts/basedrop/">basedrop</a>, which implements a memory reclamation system tailored to the constraints of real-time audio scenarios. The purpose of basedrop is to make it easy to share dynamically allocated memory with a real-time audio thread while ensuring that no allocations or deallocations happen on that thread. This is accomplished by providing a set of smart pointers (analogous to <code>Box</code> and <code>Arc</code> from the Rust standard library) which do not directly free their associated allocation when dropped, but instead automatically push it onto a lock-free queue to be collected later on another thread.</p><p>Basedrop's design has some compelling benefits: it frees you from having to write code by hand every time you want to transfer an object to another thread to be freed, and if you restrict yourself to its vocabulary of smart pointers, it eliminates the possibility of accidentally dropping an allocation on the real-time thread (a mistake which can easily remain invisible if you don't have something like <a href="https://github.com/Windfisch/rust-assert-no-alloc"><code>assert_no_alloc</code></a> to catch it). However, after talking with some developers trying to make use of basedrop in real projects, it became clear to me that these benefits come at the cost of a somewhat opinionated API, making it difficult to integrate with certain program architectures. I decided that a stripped-down version of the core linked-list queue would probably have some value, and the end result of that was the <a href="https://github.com/micahrj/llq">llq</a> crate.</p>
        ]]>
    </summary>
</entry>
<entry>
    <id>https://micahrj.github.io/posts/basedrop</id>
    <title type="html">Basedrop: A garbage collector for real-time audio in Rust</title>
    <published>2021-04-26T10:18:00-05:00</published>
    <updated>2021-04-26T10:18:00-05:00</updated>
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            <p>In real-time audio, deadlines are critical. Your code has on the order of several milliseconds to fill a buffer with samples to be shipped off to the <a href="https://en.wikipedia.org/wiki/Digital-to-analog_converter">DAC</a>, milliseconds which it may be sharing with a number of other audio plugins. If your code takes too long to produce those samples, there are no second chances; they simply won't get played, and the user will hear an objectionable glitch or stutter instead.</p><p>In order to prevent this, real-time audio code must avoid performing any operations that can block the audio thread for an unbounded or unpredictable amount of time. Such operations include file and network I/O, memory allocation and deallocation, and the use of locks to synchronize with non-audio threads; these operations are not considered &quot;real-time safe.&quot; Instead, operations like I/O and memory allocation should be performed on other threads, and synchronization should be performed using primitives that are wait-free for the audio thread. A more thorough overview of the subject can be found in Ross Bencina's now-classic blog post <a href="http://www.rossbencina.com/code/real-time-audio-programming-101-time-waits-for-nothing">&quot;Time Waits for Nothing&quot;</a>.</p><p>Given that audio software generally does need to allocate memory and make use of it from the audio thread, the question becomes how to accomplish this in a manageable and efficient way while subject to the above constraints. <a href="https://github.com/micahrj/basedrop">Basedrop</a> is my attempt at providing one answer to this question.</p><!--excerpt-->
<h3>Deferred reclamation</h3><p>Consider a simple scenario: we have a buffer of samples stored in a <code>Vec&lt;f32&gt;</code>, possibly synthesized or loaded from disk, and we would like to use it from the audio thread. As an initial sketch of a solution, we could use a wait-free bounded-capacity <a href="http://www.1024cores.net/home/lock-free-algorithms/queues">SPSC</a> channel (such as the <a href="https://crates.io/crates/rtrb"><code>rtrb</code> crate</a>) to send the buffer over to the audio thread, and then when we're done using it and want to reclaim the memory, we could send it back to a non-real-time thread over another SPSC channel to be freed.</p><p>In simple cases, this solution works well. However, it has drawbacks as an application grows in complexity. For instance, if a large number of allocations are being transferred to and from the audio thread, the fixed-capacity channel for returning allocations can fill up. Since it is not acceptable to block the audio thread in this case, the application needs to ensure either that the channel is polled frequently enough to keep up, that the channel always has worst-case capacity (using a more complex dynamically allocated design), or that the audio thread can continue without error if it is not currently possible to send back an allocation. Additionally, this solution relies on programmer discipline to ensure that allocations are always sent back to be freed, and Rust's <a href="https://en.wikipedia.org/wiki/Resource_acquisition_is_initialization">RAII</a> design makes mistakes in this regard largely invisible. Diagnostic tools like the <a href="https://crates.io/crates/assert_no_alloc"><code>assert_no_alloc</code> crate</a> can go a long way towards detecting such mistakes, but it would be nice to have a guarantee at compile time.</p><p>Basedrop's solution is to replace the fixed-capacity ring buffer for returning allocations with an <a href="http://www.1024cores.net/home/lock-free-algorithms/queues">MPSC</a> linked-list queue whose nodes are created at allocation time for (and stored inline next to) any piece of memory intended to be shared with the audio thread. When the audio thread is ready to release a piece of memory for reclamation, the corresponding node can be pushed onto the queue in an allocation-free, wait-free operation. This pattern is encapsulated by a pair of smart pointers, <code>Owned&lt;T&gt;</code> and <code>Shared&lt;T&gt;</code>, analogous to <code>Box&lt;T&gt;</code> and <code>Arc&lt;T&gt;</code>, which push their contents onto the queue for deferred reclamation rather than dropping them directly. The queue can then be processed periodically on another thread using basedrop's <code>Collector</code> type.</p><p>This system has the advantage that is impossible for the reclamation channel to become full (short of a full-on <a href="https://en.wikipedia.org/wiki/Out_of_memory">OOM</a>). It is also impossible to forget to send something back to be collected, as long as it was initially wrapped in an <code>Owned&lt;T&gt;</code> or <code>Shared&lt;T&gt;</code>. <code>Shared&lt;T&gt;</code> in particular opens up exciting possibilities for sharing immutable and persistent data structures between audio and non-audio threads in ways that would be cumbersome or impossible with the manual message-passing approach.</p><h3><code>SharedCell</code></h3><p>Basedrop provides another primitive for sharing memory with the audio thread, called <code>SharedCell&lt;T&gt;</code>. <code>SharedCell&lt;T&gt;</code> acts as a thread-safe mutable memory location for storing <code>Shared&lt;T&gt;</code> pointers, providing <code>get</code>, <code>set</code>, and <code>replace</code> methods (much like <code>Cell</code>) for fetching and updating the contents. I envision this being used as a way for a non-real-time thread to atomically publish data which can then be immutably observed by the real-time audio thread.</p><p>The main difficulty in implementing this pattern in a lock-free way lies in the fact that getting a copy of a reference-counted pointer actually consists of two steps: first, fetching the actual pointer, and then incrementing the reference count. In between these two steps, writers must not be allowed to replace the pointer with a new value, decrement the reference count for the previous value to zero, and then free its referent, as this would result in a use-after-free for the reader. There are various possible solutions to this problem with different tradeoffs.</p><p>The approach taken by <code>SharedCell&lt;T&gt;</code> is to keep a reader count alongside the stored pointer. Readers increment this count while fetching the pointer and only decrement it after successfully incrementing the pointer's reference count. Writers, in turn, after replacing the stored pointer, spin until the count is observed to be zero before they are allowed to move on and possibly decrement the reference count. This scheme is designed to be low-cost and non-blocking for readers, while being somewhat higher-overhead for writers, which I deem to be the appropriate tradeoff for real-time audio, where the reader (the audio thread) has much tighter latency deadlines and executes much more often than the writer.</p><h3>Future work</h3><p>Basedrop doesn't currently support dynamically sized types, like <code>Owned&lt;[T]&gt;</code> or <code>Owned&lt;dyn Trait&gt;</code>. This should become possible when <a href="https://doc.rust-lang.org/nightly/core/ops/trait.CoerceUnsized.html"><code>CoerceUnsized</code></a> or equivalent is stabilized. For now, it can be worked around without much issue by wrapping the DST in another layer of allocation.</p><p>Additionally, <code>Shared&lt;T&gt;</code> doesn't currently support weak references for cyclic data structures the way <code>Arc&lt;T&gt;</code> does. This would complicate the reference-counting logic (see the <a href="https://github.com/rust-lang/rust/blob/5702cfa2551a56172a4e392aab4b494562242f35/library/alloc/src/sync.rs"><code>Arc</code> source</a>), and I wanted to start with something simple that I could be sure was correct. However, this would certainly be nice to have.</p><p>I would also like to explore memory reclamation strategies with less overhead than reference counting, such as the <a href="https://www.kernel.org/doc/html/latest/RCU/whatisRCU.html">RCU</a> pattern found in the Linux kernel, <a href="https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-579.html">epoch-based reclamation</a>, and <a href="https://preshing.com/20160726/using-quiescent-states-to-reclaim-memory/">quiescent state-based reclamation</a>. I haven't yet been able to come up with a design in this vein that both dovetails with Rust ownership and satisfies the constraints of real-time audio (and audio plugins), but I think it's a promising direction for the future.</p><h3>Finally</h3><p>Basedrop is available on <a href="https://crates.io/crates/basedrop">crates.io</a>! Please feel free to give it a try in your own projects. Feedback and bug reports are welcome.</p><p>Lastly, I would like to thank William Light for some very helpful conversations while I was working out the design of basedrop.</p>
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            <p>In real-time audio, deadlines are critical. Your code has on the order of several milliseconds to fill a buffer with samples to be shipped off to the <a href="https://en.wikipedia.org/wiki/Digital-to-analog_converter">DAC</a>, milliseconds which it may be sharing with a number of other audio plugins. If your code takes too long to produce those samples, there are no second chances; they simply won't get played, and the user will hear an objectionable glitch or stutter instead.</p><p>In order to prevent this, real-time audio code must avoid performing any operations that can block the audio thread for an unbounded or unpredictable amount of time. Such operations include file and network I/O, memory allocation and deallocation, and the use of locks to synchronize with non-audio threads; these operations are not considered &quot;real-time safe.&quot; Instead, operations like I/O and memory allocation should be performed on other threads, and synchronization should be performed using primitives that are wait-free for the audio thread. A more thorough overview of the subject can be found in Ross Bencina's now-classic blog post <a href="http://www.rossbencina.com/code/real-time-audio-programming-101-time-waits-for-nothing">&quot;Time Waits for Nothing&quot;</a>.</p><p>Given that audio software generally does need to allocate memory and make use of it from the audio thread, the question becomes how to accomplish this in a manageable and efficient way while subject to the above constraints. <a href="https://github.com/micahrj/basedrop">Basedrop</a> is my attempt at providing one answer to this question.</p>
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<entry>
    <id>https://micahrj.github.io/posts/amplitwist</id>
    <title type="html">The amplitwist, the conjugate transpose, and the complex derivative</title>
    <published>2019-12-29T14:50:44-06:00</published>
    <updated>2019-12-29T14:50:44-06:00</updated>
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            <p>Complex numbers have a representation as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> matrices, which can serve to illuminate some initially non-obvious aspects of how they work. A real number <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> can be represented as a multiple of the identity matrix:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>with addition and multiplication given by the corresponding matrix operations. In order to extend this representation to the complex numbers, we need a matrix <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.604em;vertical-align:-0.95em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.654em;"><span style="top:-3.9029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>We can thus represent any complex number <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">bi</span></span></span></span> as:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>It can be verified that addition and multiplication of these matrices is equivalent to addition and multiplication of the complex numbers they represent (meaning that matrices of this form comprise a field isomorphic to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>).</p><!--excerpt-->
<h3>The amplitwist</h3><p>Just as any complex number <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">bi</span></span></span></span> can be written in polar form <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span></span></span></span></span></span>, a matrix of the above form can be written as a scaled rotation (or an “amplitwist,” as Tristan Needham refers to it in <a href="http://usf.usfca.edu/vca/"><em>Visual Complex Analysis</em></a>):</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>(This is a special case of the more general <a href="https://en.wikipedia.org/wiki/Polar_decomposition">polar decomposition</a> for matrices, by which a square matrix can be written as the product of a symmetric positive-definite matrix and an orthogonal matrix.)</p><p>In fact, these matrices act on vectors in the plane in the same way that complex numbers act on one another: by scaling and rotation. So, we can look at complex multiplication as a particular binary operation on vectors in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, or we can look at it as standard matrix multiplication on a particular class of matrices in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mbin mtight">×</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>, <em>or</em> we can look at it as multiplication of vectors in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> by that particular class of matrices.</p><h3>The conjugate transpose</h3><p>When moving from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>, the proper generalizations of many constructions from linear algebra involve the complex conjugate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9778em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span></span></span><span style="top:-3.8144em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">bi</span></span></span></span> and the conjugate transpose <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1223em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1223em;"><span style="top:-3.3362em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">T</span></span></span></span></span></span></span></span></span></span></span>:</p><ul><li>The real inner product <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> generalizes to the complex inner product <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></li><li>Symmetric matrices <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">T</span></span></span></span></span></span></span></span></span></span></span> generalize to Hermitian matrices <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></li><li>Orthogonal matrices <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">T</span></span></span></span></span></span></span></span></span></span></span> generalize to unitary matrices <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></li></ul><p>and so on. There are various explanations for this. One that I am fond of involves replacing the individual complex elements in a matrix or vector with their <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> matrix representations, turning a complex column vector into a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> real block matrix:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.26em;vertical-align:-1.88em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.667em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.200em" viewBox="0 0 667 4200"><path d="M403 1759 V84 H666 V0 H319 V1759 v600 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v600 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.88em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.667em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.200em" viewBox="0 0 667 4200"><path d="M347 1759 V0 H0 V84 H263 V1759 v600 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v600 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.08em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.55em;"><span style="top:-5.55em;"><span class="pstrut" style="height:8.6em;"></span><span style="width:0.667em;height:6.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="6.600em" viewBox="0 0 667 6600"><path d="M403 1759 V84 H666 V0 H319 V1759 v3000 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v3000 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-0.9675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">d</span></span></span><span style="top:-0.9675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.55em;"><span style="top:-5.55em;"><span class="pstrut" style="height:8.6em;"></span><span style="width:0.667em;height:6.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="6.600em" viewBox="0 0 667 6600"><path d="M347 1759 V0 H0 V84 H263 V1759 v3000 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v3000 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>and a complex matrix into a <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span></span></span></span> real block matrix:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.26em;vertical-align:-1.88em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.667em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.200em" viewBox="0 0 667 4200"><path d="M403 1759 V84 H666 V0 H319 V1759 v600 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v600 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.88em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.04em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.18em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.98em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.88em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">hi</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.88em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.667em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.200em" viewBox="0 0 667 4200"><path d="M347 1759 V0 H0 V84 H263 V1759 v600 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v600 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span></span></p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.08em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.55em;"><span style="top:-5.55em;"><span class="pstrut" style="height:8.6em;"></span><span style="width:0.667em;height:6.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="6.600em" viewBox="0 0 667 6600"><path d="M403 1759 V84 H666 V0 H319 V1759 v3000 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v3000 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">e</span></span></span><span style="top:-0.9675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span><span style="top:-0.9675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.24em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.04em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.18em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.98em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-0.78em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span><span style="top:-0.9675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.58em;"><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">d</span></span></span><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.1675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">h</span></span></span><span style="top:-0.9675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.08em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.55em;"><span style="top:-5.55em;"><span class="pstrut" style="height:8.6em;"></span><span style="width:0.667em;height:6.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="6.600em" viewBox="0 0 667 6600"><path d="M347 1759 V0 H0 V84 H263 V1759 v3000 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v3000 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.05em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>Since the transpose of an individual <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> block</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.639em;vertical-align:-0.95em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.689em;"><span style="top:-3.9029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">T</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>corresponds to the conjugate of the original complex number, the original notions of inner product, symmetric matrix, orthogonal matrix, and so on give the same results over these block matrices as their complex generalizations do over complex vectors and matrices.</p><h3>The complex derivative</h3><p>A complex function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> can be looked at as a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. The conditions for continuity are the same. However, the conditions for differentiability are different.</p><p>The derivative of a real function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span></span></span></span> at a point <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> is the linear function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> which best locally approximates <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> at that point. It can be written as the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> Jacobian matrix of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>'s partial derivatives:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.5038em;vertical-align:-2.0019em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.667em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.200em" viewBox="0 0 667 4200"><path d="M403 1759 V84 H666 V0 H319 V1759 v600 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v600 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5019em;"><span style="top:-4.5019em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0019em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5019em;"><span style="top:-4.5019em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.3714em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0019em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.667em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="4.200em" viewBox="0 0 667 4200"><path d="M347 1759 V0 H0 V84 H263 V1759 v600 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v600 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>All that is necessary for such a function to be differentiable is for each of these partial derivatives to exist. If <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> is instead considered as a complex function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord mathnormal">i</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">i</span></span></span></span>, its derivative at a point <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord mathnormal">i</span></span></span></span> should again be the best local linear approximation, but this time it should be a linear function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> of a single complex variable, meaning that it can be expressed as a single complex number to be multiplied by its argument.</p><p>Which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> Jacobian matrices can we pack into a single complex number? In other words, which <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> real matrices act on a vector in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> the way complex numbers act on one other? We discovered this above: they are the matrices of the form</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>i.e. scaled rotation matrices or amplitwists. So a function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord mathnormal">i</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">i</span></span></span></span> is differentiable if and only if the following conditions hold:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>These are known as the Cauchy-Riemann equations.</p><p>Functions with this property are known as holomorphic. This turns out to be a much stronger condition than differentiability over <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, with correspondingly much stronger implications:</p><ul><li>Holomorphic functions are <a href="https://en.wikipedia.org/wiki/Analytic_function">analytic</a>, i.e. they are everywhere locally equal to their Taylor series</li><li>Both the real and imaginary parts of a holomorphic function are <a href="https://en.wikipedia.org/wiki/Harmonic_function%22">harmonic</a>, i.e. their Laplacian vanishes everywhere</li><li>A holomorphic function is <a href="https://en.wikipedia.org/wiki/Conformal_map">conformal</a>, i.e. it locally preserves angles, as long as its derivative is nonzero everywhere</li></ul>
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    <summary type="html">
        <![CDATA[
            <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
            <p>Complex numbers have a representation as <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> matrices, which can serve to illuminate some initially non-obvious aspects of how they work. A real number <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> can be represented as a multiple of the identity matrix:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>with addition and multiplication given by the corresponding matrix operations. In order to extend this representation to the complex numbers, we need a matrix <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span></span></span></span> such that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.604em;vertical-align:-0.95em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.654em;"><span style="top:-3.9029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>We can thus represent any complex number <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">bi</span></span></span></span> as:</p><p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>It can be verified that addition and multiplication of these matrices is equivalent to addition and multiplication of the complex numbers they represent (meaning that matrices of this form comprise a field isomorphic to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>).</p>
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</entry>
<entry>
    <id>https://micahrj.github.io/posts/a-rotated-video-of-a-cube</id>
    <title type="html">A rotated video of a cube</title>
    <published>2019-01-07T21:52:00-06:00</published>
    <updated>2019-01-07T21:52:00-06:00</updated>
    <link rel="alternate" href="https://micahrj.github.io/posts/a-rotated-video-of-a-cube" type="text/html"/>
    <content type="html">
        <![CDATA[
            <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
            <p>The following is a video of a rotating cube:</p><p><video autoplay loop muted src="/posts/a-rotated-video-of-a-cube/cube.mp4" type="video/mp4"></video></p><p>This can be visualized as a plane sweeping through a cube of spacetime, or a flipbook:</p><p><video autoplay loop muted src="/posts/a-rotated-video-of-a-cube/scan.mp4" type="video/mp4"></video></p><!--excerpt-->
<p>If the plane is instead swept as follows,</p><p><video autoplay loop muted src="/posts/a-rotated-video-of-a-cube/scan-sideways.mp4" type="video/mp4"></video></p><p>the resulting video is:</p><p><video autoplay loop muted src="/posts/a-rotated-video-of-a-cube/cube-sideways.mp4" type="video/mp4"></video></p>
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    </content>
    <summary type="html">
        <![CDATA[
            <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
            <p>The following is a video of a rotating cube:</p><p><video autoplay loop muted src="/posts/a-rotated-video-of-a-cube/cube.mp4" type="video/mp4"></video></p><p>This can be visualized as a plane sweeping through a cube of spacetime, or a flipbook:</p><p><video autoplay loop muted src="/posts/a-rotated-video-of-a-cube/scan.mp4" type="video/mp4"></video></p>
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</entry>
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