On the Quadraticity of Linear Combinations of Quadratic Matrices

Abstract

Let [GRAPHICS] and [GRAPHICS] be [GRAPHICS] nonzero quadratic matrices and let [GRAPHICS] and [GRAPHICS] be nonzero complex numbers. Necessary and sufficient conditions for the quadraticity of the linear combinations of the form [GRAPHICS] are obtained. Our main results contain many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices. Finally, some numerical examples are given to exemplify the main results.