Euclidean-Based Approaches for Solving Coprime Integer Matrices

Abstract

Coprime pairs of integer matrices have applications in various fields, including multirate systems and multidimensional signal processing. This article addresses problems related to coprime matrices when given two integer matrices of the same size. These problems include determining whether the matrices are coprime, computing their greatest common divisor (gcd), and finding two unknown integer matrices that satisfy Bezout’s equation. First, we present a division theorem for two matrices, which guarantees that the remainder matrix is either the zero matrix or a nonsingular matrix with a smaller absolute determinant than that of the divisor. The gcd of two integer matrices can be computed by repeatedly applying the division theorem until the remainder matrix becomes zero. We then propose an algorithm for finding the gcd and another for determining Bezout’s coefficient matrices from a given pair of integer matrices. Finally, we provide the general solution to Bezout’s equation for nonsingular and commuting integer matrices. This article offers a theoretical derivation of a method for solving Bezout’s equation for integer matrices, generalizing the Euclidean algorithm for integers to integer matrices. Simulations demonstrate that these approaches significantly improve computational efficiency.