Null ideals of sets of 3 × 3 similar matrices with irreducible characteristic polynomial

Abstract

Let F be a field and the ring of matrices over F. Given a subset S of , the null ideal of S is the set of all polynomials f with coefficients from such that for all . We say that S is core if the null ideal of S is a two-sided ideal of the polynomial ring . We study sufficient conditions under which S is core in the case where S consists of matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if F is finite with q elements and , then S is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.