On some matrix diophantine equations

Abstract

Let $A \in M_n(\mathbf{C})$, $n \ge 2$ be the matrix which has at least one real eigenvalue $\alpha \in (0, 1)$. If the matrix equation \begin{equation} A^x + A^y + A^z = A^w \tag{1} \end{equation} is satisfied in positive integers $x$, $y$, $z$, $w$, then $\max \{x-w, y-w, z-w\} \ge 1$. If suppose that the matrix $A$ has at least one real eigenvalue $\alpha > \sqrt{2}$ and the equation (1) is satisfied in positive integers $x$, $y$, $z$ and $w$, then $\max \{x-w, y-w, z-w\} = -1$. Moveover, we investigate the solvability of the matrix equations (1) and \begin{equation} A^x + A^y = A^z \tag{2} \end{equation} for the non-negative real $n \times n$ matrices, where $|\det A| > 1$, in positive integers $x$, $y$, $z$, $w$ for (1) and $x$, $y$, $z$ for (2). Using the wellknown theorem of Perron-Frobenius we obtain some informations concerning solvability these equations.