On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix

Abstract

We characterize the involutiveness of the linear combinations of the form $a{\mathbf{A}} + b{\mathbf{B}}$ when $a,b$ are nonzero complex numbers, ${\mathbf{A}}$ is a quadratic $n \times n$ nonzero matrix and ${\mathbf{B}}$ is an arbitrary $n \times n$ nonzero matrix, under certain properties imposed on $\mathbf{A}$ and $\mathbf{B}$.