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For instance, is there some way I can say "let A and B be arbitrary real $m\times n$ and $k\times m$ matrices, Simplify[Transpose[Transpose[A].Transpose[B]]]" and Mathematica would simplify it to B.A?

I know I can set A and B to be matrices containing symbols (e.g. A = Table[Subscript[a,i,j],{i,m},{j,n}]), but results can get quite messy if the problem is more complex than Transpose[Transpose[A].Transpose[B]]

EDIT: To answer @Searke and @Artes questions in the comments: I'm currently watching this Stanford online machine learning course. If you look at the lecture notes, pages 8-11, you see a some matrix calculations. I can follow these calculations with pen and paper, but I haven't found a way to derive e.g. this result from page 11 using Mathematica:

enter image description here

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Nope. The issue is that for a given symbol there is no way to say "Oh this symbol is a symmetric, real matrix." To the best of my knowledge, there is no package for this. – Searke Mar 20 '12 at 13:17
I would actually be very interested in hearing what people think such functionality should be able to do. Does some other software do this and how do they do it? – Searke Mar 20 '12 at 13:18
See related question:… – Eli Lansey Oct 4 '12 at 11:16
What about this answer? – chris Apr 20 '14 at 20:31
up vote 17 down vote accepted

Initially, Mathematica is not designed for such abstract calculations.

But, Mathematica is a powerful programming language, so that one can add such functionality easily.

See the following examples in related area of differential geometry:

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I guess that V9 now adds this capability:

$Assumptions = {
  Element[A, Matrices[{m, n}]],
  Element[B, Matrices[{n, k}]]

(* Out: B.A *)
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I am not sure, but maybe this software for Mathematica could somehow help. The software is for a package called NCAlgebra developed by UC San Diego. I am not familiar with the detailed usage, but it claims to implement capability to study noncommutative inequalities, linear controls, and semidefinite programming within Mathmeatica.

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Hi Konstantin, welcome to Mathematica.SE. Can you add some information about the software here? When that link dies ("when", not "if") your answer becomes useless. – stevenvh Oct 3 '12 at 17:56

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