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So I'm wondering, first off, where I can read up to get a feel for state-of-the-art matrix multiplication concepts. I'll try to be more specific:

I'm wondering if there has been research on circuits that compute a pre-defined matrix function. For instance, if we have a function such as $(AB)+C+(DEF)(AB)$, where all the variables are matrices with given sizes, how efficient can we make a circuit that computes this function? I've been toying with an idea in which the entries are all computed modulo $p$. So I'm wondering, broadly, what I can read up on to be up-to-date.

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While I don't know about the complexity of matrix polynomials in general, the state of the art in the complexity of matrix multiplication is nicely summarized in a recent ToC Graduate Survey by Markus Bläser.

Computing entries mod $p$ is basically the same as the question of the complexity of performing the computation over the field $\mathbb{F}_p$, which is fairly well-studied (though perhaps not as well-studied as characteristic zero).

Most matrix multiplication research works in the nonuniform setting anyways, so it should almost all be relevant to the question of circuit size.

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