On Characterization of Being a Matrix Q (k) g(a2,b2) of Linear Combinations of a Matrix Q (n) g(a1,b1) and a Matrix Q m

Abstract

It is given a characterization of all solution of the matrix equation $c_{1}Q_{g(a_{1}, b_{1})}^{(n)}+c_{2}Q^{m}=Q_{g(a_{2}, b_{2})}^{(k)}$ with unknowns $c_{1}, c_{2} \in \mathbb{C}^{*}$. Here the matrix $Q_{g(a, b)}^{(l)}$, called an $l$-generalized Fibonacci $Q$-matrix, is defined by means of the Fibonacci $Q$-matrix, where $l$ is an integer, and $a, b \in \mathbb{R}^{*}$.