FINITE MATRICES SIMILAR TO IRREDUCIBLE ONES

Abstract

In this paper, we prove that an $n\times n~(n\geq 3)$ complex matrix $T$ is similar to an irreducible matrix if and only if $T$ is not quadratic and rank $(T-\lambda I)\geq n/2$ for every complex number $\lambda$. As an application, we prove that: for any integers $n$ and $k$ with $ 3 \leq k \lt n$, there exists an $n \times n$ irreducible nilpotent matrix of index $k$. This answers a question posed by P. R. Halmos