An elementary method to compute the algebra generated by some given matrices and its dimension

TL;DR

The paper presents an efficient method to determine if a matrix belongs to the algebra generated by given matrices and to compute the algebra's dimension, simplifying analysis in linear algebra contexts.

Contribution

It introduces a straightforward approach for identifying membership and calculating the dimension of the algebra generated by specific matrices.

Findings

Provides an efficient algorithm for algebra membership testing.

Offers a simple method to compute the dimension of generated algebras.

Enhances computational tools for matrix algebra analysis.

Abstract

We give an efficient solution to the following problem: Given $X_1, \ldots X_d$ and $Y$ some $n$ by $n$ matrices can we determine if $Y$ is in the unital algebra generated by $X_1, \ldots, X_d$ as a subalgebra of all $n$ by $n$ matrices? The solution also gives an easy method for computing the dimension of this algebra.