Irreducibility criterion for the set of two matrices

TL;DR

This paper establishes criteria for irreducibility, Schur irreducibility, and indecomposability of pairs of matrices, linking algebraic properties to graph structures and classifying maximal subalgebras of matrix algebras.

Contribution

It provides explicit criteria for matrix set irreducibility and classifies maximal subalgebras of matrix algebras using graph-theoretic methods.

Findings

Criteria for irreducibility, Schur irreducibility, and indecomposability of matrix pairs.

Complete classification of maximal subalgebras of ${ m Mat}(n,{ m C})$.

Description of minimal generating sets of elementary matrices for the full matrix algebra.

Abstract

We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two $n\times n$ matrices $\Lambda_n$ and $A_n$ in terms of the subalgebra associated with the "support" of the matrix $A_n$, where $\Lambda_n$ is a diagonal matrix with different non zeros eigenvalues and $A_n$ is an arbitrary one. The list of all maximal subalgebras of the algebra ${\rm Mat}(n,{\mathbb C})$ and the list of the corresponding invariant subspaces connected with these two matrices is also given. The properties of the corresponding subalgebras are expressed in terms of the graphs associated with the support of the second matrix. For arbitrary $n$ we describe all minimal subsets of the elementary matrices $E_{km}$ that generate the algebra ${\rm Mat}(n,{\mathbb C})$.