Powers of matrices with positive definite real part

Abstract:

For $n$ by $n$ complex matrices $A$ the following two facts are proven by elementary techniques: 1. If ${A^m}$ is never normal, $m \in {I^ + }$, then the equation $x{A^m}{x^\ast } = 0$ has a solution $0 \ne x \in {C^n},m \in {I^ + }$; 2. If $H(A) = (A + {A^\ast })/2$ is positive definite, then $H({A^m})$ is positive definite for all $m \in {I^ + }$ if and only if $A$ is Hermitian.