Finite matrices similar to irreducible ones

Abstract

In this paper, we prove that an n x n (n greater than or equal to 3) complex matrix T is similar to an irreducible matrix if and only if T is not quadratic and rank (T - lambdaI) greater than or equal to n/2 for every complex number lambda. As an application, we prove that: for any integers n and k with 3 less than or equal to k < n, there exists an n x n irreducible nilpotent matrix of index k. This answers a question posed by P. R. Halmos.