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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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    $\begingroup$ 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. $\endgroup$
    – Searke
    Commented Mar 20, 2012 at 13:17
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    $\begingroup$ 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? $\endgroup$
    – Searke
    Commented Mar 20, 2012 at 13:18
  • $\begingroup$ See related question: mathematica.stackexchange.com/questions/8/… $\endgroup$
    – Eli Lansey
    Commented Oct 4, 2012 at 11:16
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    $\begingroup$ What about this answer? mathematica.stackexchange.com/a/16378/1089 $\endgroup$
    – chris
    Commented Apr 20, 2014 at 20:31
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    $\begingroup$ There's a MatrixD package that lets you differentiate matrix expressions -- mathematica.stackexchange.com/questions/138708/… $\endgroup$
    – Yaroslav Bulatov
    Commented Apr 26, 2017 at 19:49

4 Answers 4

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For the posted example, TensorReduce does the trick:

TensorReduce[
  Transpose[Transpose[A].Transpose[B]], 
  Assumptions -> {A ∈ Matrices[{m, n}], B ∈ Matrices[{k, m}]}
]

B.A
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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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  • $\begingroup$ Links are dead. $\endgroup$
    – jlh
    Commented Mar 26, 2020 at 8:58
  • $\begingroup$ @jlh, that vendor seems to be no more, but I've added links to the Wayback Machine. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented May 12, 2020 at 11:46
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Indeed this is a one liner in NCAlgebra:

<< NC`
<< NCAlgebra`
NCGrad[1/2 (x ** z - y)^T ** (x ** z - y), z]

which results in

-y^T ** x + z^T ** x^T ** x

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I am not sure, but maybe this software for Mathematica http://www.math.ucsd.edu/~ncalg/ 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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    $\begingroup$ 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. $\endgroup$
    – stevenvh
    Commented Oct 3, 2012 at 17:56

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