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

Let F be a field and $ M_n(F) $ M n (F) the ring of $ n \times n $ n × n matrices over F. Given a subset S of $ M_n(F) $ M n (F) , the null ideal of S is the set of all polynomials f with coefficients from $ M_n(F) $ M n (F) such that $ f(A) = 0 $ f (A) = 0 for all $ A \in S $ A ∈ S. We say that S is core if the null ideal of S is a two-sided ideal of the polynomial ring $ M_n(F)[x] $ M n (F) [ x ]. We study sufficient conditions under which S is core in the case where S consists of $ 3 \times 3 $ 3 × 3 matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if F is finite with q elements and $ |S| \ge q^3-q^2+1 $ | S | ≥ q 3 − q 2 + 1 , 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. [ABSTRACT FROM AUTHOR]