On some applications of a particular class of block matrices

Abstract

In this paper, we study a particular class of block matrices placing an emphasis on their spectral properties. Some related applications are then presented. In particular, we prove the existence of an infinite number of integer matrix solutions for the two equations: aX p + bY p = cZ p and aX p + bY q = cZ r for any integers a, b and c and for any positive integers p, q and r such that these solution matrices have all of their entries nonzero natural numbers.