# Matrix algebra

Maybe also some operator algebra

July 9, 2018 — March 11, 2024

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

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

This page mostly exists to bookmark tools that I have found useful to do abstract 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. 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

TBC

## 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 provides 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

*Matrix Analysis*. Graduate Texts in Mathematics.

*Positive definite matrices*.

*Linear Algebra and Its Applications*.

*Advances in Automatic Differentiation*.

*Matrix Computations*.

*Old and new matrix algebra useful for statistics*.

*Theoretical Computer Science*.

*Wiley StatsRef: Statistics Reference Online*.

*Matrix Algebra Useful for Statistics*.

*A Matrix Handbook for Statisticians*.

*SIAM Review*.

*Matrix Concentration & Computational Linear Algebra / ENS Short Course*.

*SIAM Journal on Mathematics of Data Science*.