A General Approach to Coprime Pairs of Matrices, Based on Minors

In signal processing, coprime pairs of integer matrices have an important role in the study of multidimensional multirate systems, and in multidimensional array processing. In this paper a general theorem for coprimality of pairs of matrices is first presented, based on the minors of an associated composite matrix. First, a necessary and sufficient set of conditions for coprimality is presented based on minors. This result applies to matrices of any size. Several useful consequences of the result are discussed, and it is first applied to the 2 × 2 case, including special cases such as Toeplitz matrices, commuting Toeplitz matrices, circulants, and skew circulants. The condition under which a 2 × 2 integer matrix and its ajdugate are coprime is also derived. The minor based result is then applied extensively for the 3 × 3 case to derive new coprimality conditions for several matrix pairs. In particular, adjugates of matrices are considered in detail because adjugates always commute with the parent matrices. The so-called generalized 3 × 3 circulant matrix and its adjugate are then analyzed in detail. Generalized circulants also form a commuting family. Special cases such as 3 × 3 circulant pairs and skew circulant pairs are also elaborated.