Common Invariant Subspace and Commuting Matrices

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.