Matrix calculus

We can generalise the high school calculus, which is about scalar functions of a scalar argument, in various ways, to handle matrix-valued functions or matrix-valued arguments. One could generalise this further, by to full tensor calculus. But it happens that specifically matrix/vector operations are at a useful point of complexity for lots of algorithms, kind of a MVP. (I usually want this for higher order gradient descent.)

I will mention two convenient and popular formalisms for doing that here. In practice a mix of each is often useful.

Matrix differentials

🏗 I need to return to this and tidy it up with some examples.

A special case of tensor calculus that happens to be handy for some common cases; where the rank of the argument and value of the function is not too big. Fun pain point: agreeing upon layout of derivatives, numerator vs denominator.

If our problem is nice, this often gets us a low-fuss, compact, tidy solution even for some surprising cases where it seems that more general tensors would be more natural. (for which, see below)

Indexed tensor calculus

Filed under multilinear algebra.


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Turkington, Darrell A. 2001. Matrix Calculus Zero-One Matrices. Cambridge University Press.

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