Finite matrices similar to irreducible ones (Q1590402)

An \(n\)-square matrix \(T\) is said to be irreducible if it commutes with no (orthogonal) projection other than \(O\) and \(I,\) and it is said to be quadratic if there exist complex numbers \(\alpha, \beta\) such that \(T^2 + \alpha T + \beta I = O.\) The following result is proved: An \(n\)-square matrix \(T\) (\(n \geq 3\)) is similar to an irreducible matrix if and only if \(T\) is not quadratic and \(\text{ rank}(T - \lambda I) \geq \frac{n}{2}\) for every complex number \(\lambda.\) This answers a question posed by \textit{P. R. Halmos} [Linear algebra problem book (1995; Zbl 0846.15001)].