Can Mathematica do symbolic linear algebra?

Question

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

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: http://mathematica.stackexchange.com/questions/8/how-to-symbolically-do-matrix-block-inversion — Eli Lansey, Oct 04 '12 at 11:16
What about this answer? http://mathematica.stackexchange.com/a/16378/1089 — chris, Apr 20 '14 at 20:31
There's a MatrixD package that lets you differentiate matrix expressions -- https://mathematica.stackexchange.com/questions/138708/differentiating-functions-of-vectors-matrices/141237#141237 — Yaroslav Bulatov, Apr 26 '17 at 19:49

score 28 · Answer 1 · edited May 12 '20 at 11:46

28

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:

calculations in symbolic dimensions
Abstract calculations

edited May 12 '20 at 11:46

J. M.'s missing motivation

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answered Mar 20 '12 at 15:47

helen

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Links are dead. – jlh Mar 26 '20 at 08:58
@jlh, that vendor seems to be no more, but I've added links to the Wayback Machine. – J. M.'s missing motivation May 12 '20 at 11:46

Mauricio de Oliveira · Answer 2 · 2017-04-26T19:12:15.597

19

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

edited Apr 26 '17 at 19:12

answered Apr 19 '17 at 04:06

Mauricio de Oliveira

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score 12 · Answer 3 · edited Apr 22 '14 at 01:32

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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.

edited Apr 22 '14 at 01:32

Carl Morris

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answered Oct 03 '12 at 17:47

Konstantin

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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 03 '12 at 17:56

score 12 · Accepted Answer · answered May 12 '20 at 21:04

12

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

answered May 12 '20 at 21:04

Greg Hurst

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Can Mathematica do symbolic linear algebra?

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