Common Invariant Subspace and Commuting Matrices

TL;DR

This paper investigates conditions under which matrices with a common invariant subspace commute, especially over extension fields, and applies these findings to solve matrix equations.

Contribution

It establishes new criteria for matrix commutativity based on invariant subspaces and Galois group properties, extending understanding of matrix structure over fields.

Findings

Matrices with a common invariant subspace of dimension k commute if 1<k<n-1.

If the Galois group is symmetric or alternating, matrices commute under certain conditions.

For prime n, matrices with a common invariant subspace commute, except in specific cases when n=4.

Abstract

Let $K$ be a perfect field, $L$ be an extension field of $K$ and $A,B\in\mathcal{M}_n(K)$. If $A$ has $n$ distinct eigenvalues in $L$ that are explicitly known, then we can check if $A,B$ are simultaneously triangularizable over $L$. Now we assume that $A,B$ have a common invariant proper vector subspace of dimension $k$ over an extension field of $K$ and that $\chi_A$, the characteristic polynomial of $A$, is irreducible over $K$. Let $G$ be the Galois group of $\chi_A$. We show the following results i) If $k\in{1,n-1}$, then $A,B$ commute. ii) If $1\leq k\leq n-1$ and $G=\mathcal{S}_n$ or $G=\mathcal{A}_n$, then $AB=BA$. iii) If $1\leq k\leq n-1$ and $n$ is a prime number, then $AB=BA$. Yet, when $n=4,k=2$, we show that $A,B$ do not necessarily commute if $G$ is not $\mathcal{S}_4$ or $\mathcal{A}_4$. Finally we apply the previous results to solving a matrix equation.