Irreducibility criterion for the set of two matrices

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 ${\rm Mat}(n,{\mathbb C})$.