Common invariant subspace and commuting matrices

Abstract

Let K be a perfect field, L be an extension field of K and A,B∈Mn(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 χA, the characteristic polynomial of A, is irreducible over K. Let G be the Galois group of χA. We show the following results(i)If k∈{1,n-1}, then A,B commute.(ii)If 1⩽k⩽n-1 and G=Sn or G=An, then AB=BA.(iii)If 1⩽k⩽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 S4 or A4. Finally we apply the previous results to solving a matrix equation.