A Construction of Pairwise Co-Prime Integer Matrices of Any Dimension and Their Least Common Right Multiple

Abstract

Compared with co-prime integers, co-prime integer matrices are more challenging due to the non-commutativity. In this paper, we present a new family of pairwise co-prime integer matrices of any dimension and large size. These matrices are non-commutative and have low spread, i.e., their ratios of peak absolute values to mean absolute values (or the smallest non-zero absolute values) of their components are low. When matrix dimension is larger than 2, this family of matrices differs from the existing families, such as circulant, Toeplitz matrices, or triangular matrices, and therefore, offers more varieties in applications. In this paper, we first prove the pairwise coprimality of the constructed matrices, then determine their determinant absolute values, and their least common right multiple (lcrm) with a closed and simple form. We also analyze their sampling rates when these matrices are used as sampling matrices for a multi-dimensional signal. The proposed family of pairwise co-prime integer matrices may have applications in multi-dimensional Chinese remainder theorem (MD-CRT) that can be used to determine integer vectors from their integer vector remainders modulo a set of integer matrix moduli, and also in multi-dimensional sparse sensing and multirate systems.