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Is it possible to get widely available math software (Maple/Matlab/Mathematica, etc) to symbolically differentiate vector and scalar functions of matrices, returning the result in terms of the original matrices and vectors involved? I have in mind the simple sort of rules collected here for example.

On a few separate occasions I've scoured around the internet for such a thing and only turned up a bunch of incomplete threads of various vintage (like this or this or this).

So my main question is if I am missing the right keywords to find what is obvious to people who use this sort of functionality all the time, and what platform it is available on if so.

If in fact this sort of functionality is not available in any of the commonly used software my question is if this is because of some sort of practical obstruction I am not seeing or simply because the problems for which it would be useful are simple enough to be done by hand (which is what I've ended up doing after I spend 4 hours searching for the "easy" way).

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You might find the following software useful: – Suvrit Mar 2 '12 at 5:33
I would also be interested in a package implementing basic matrix algebra only --- you know, dealing with non-commuting symbols and applying automatically simple stuff like $(AB)^*=B^*A^*$. Is there a good one around? – Federico Poloni Mar 2 '12 at 8:19
It seems as though Mathematica has some functionality for this: I don't know about general functions though, but it seems like it can handle it. – Pádraig Ó Conbhuí Mar 26 '12 at 3:50

I have not used it, but the NCAlgebra (NC for noncommutative) extension for Mathematica does do directional derivatives at least. It is aimed mostly at algebra involving noncommutative variables and their adjoints, however, so that may or may not be sufficient for your needs.

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