Matrix algebra

Maybe also some operator algebra

July 9, 2018 — March 11, 2024

functional analysis
linear algebra
Figure 1

Algebra over matrices, which are the things that define the linear operators that we care about when we definte operators over vectors.

If we admit infinitesimal matrices, then this gets us matrix calculus, and various nice matrix representaions, matrix inverses and such.

This page mostly exists to bookmark tools that I have found useful to do abstracts matrix algebra, e.g. without pre-committing to a size or whatever.

This is not wildly esoteric or difficult, but there is a lot of fiddly book-keeping, so a symbolic mathematics package helps. An unfortunately, those packages have wildly esoteric and difficult documentation to understand. This page is mostly to bookmark those packages and relevant manual pages.

I may yet put some other results in though.

1 Ore algebras

Look interesting. TBC

2 Gröbner bases


3 Tooling

The keyword that we are looking for is non-commutative algebra, since matrix algebras are non-commutative. There are many non-commutative algebras and most of them are more complicated than the matrix ones.

3.1 Mathematica

Lots of handy extensions for non-commutative algebras in general. See NCAlgebra et al. If you have a license, start here.

3.2 SymPy

SymPy has a non-commutative algebra package, see Quantum Mechanics, which provide operators over complex fields. Restricting ourselves to the reals probably gets us what we want.

3.3 Sage

I think that this is probably powerful, but I have gotten rather lost in the documentation.

We can go bareback and simply define Algebras with the commutative=False option, I think. Then what do we do?

Noncommutative Algebras in Sage introduces several ways of solving problems, and notes that they possibly all depend upon PLURAL.

Specific algebra that might be of interest:

mkauers/ore_algebra supports Ore Algebra operations in Sage. Maybe that does what I want?

4 References

Bhardwaj, Klep, and Magron. 2021. Noncommutative Polynomial Optimization.”
Bhatia. 1997. Matrix Analysis. Graduate Texts in Mathematics.
———. 2015. Positive definite matrices.
de Oliveira. 2012. Simplification of Symbolic Polynomials on Non-Commutative Variables.” Linear Algebra and Its Applications.
Giles. 2008. Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation.” In Advances in Automatic Differentiation.
Golub, and van Loan. 1983. Matrix Computations.
Helton. 2017. The NCAlgebra Suite - Version 5.0.”
Minka. 2000. Old and new matrix algebra useful for statistics.
Mora. 1994. An Introduction to Commutative and Noncommutative Gröbner Bases.” Theoretical Computer Science.
Petersen, and Pedersen. 2012. The Matrix Cookbook.”
Searle. 2014. Matrix Algebra.” In Wiley StatsRef: Statistics Reference Online.
Searle, and Khuri. 2017. Matrix Algebra Useful for Statistics.
Seber. 2007. A Matrix Handbook for Statisticians.
Simoncini. 2016. Computational Methods for Linear Matrix Equations.” SIAM Review.
Tropp. 2019. Matrix Concentration & Computational Linear Algebra / ENS Short Course.
Yurtsever, Tropp, Fercoq, et al. 2021. Scalable Semidefinite Programming.” SIAM Journal on Mathematics of Data Science.