Coprimality of Certain Families of Integer Matrices

Commuting coprime pairs of integer matrices have been of interest in multidimensional multirate systems, and more recently in array processing. In multirate systems they arise, for example, in the design of interchangeable cascades of decimator and expander matrices. In array processing they arise in the construction of dense coarrays from sparse sensors located on a pair of lattices. For the important case of two dimensional signals, these matrices have size 2 × 2. In this paper the condition for coprimality is derived for several classes of 2 × 2 integer matrices, namely circulant, skew-circulant, and triangular families. The first two are also commuting families. For each class, the special case of adjugate pairs, which automatically commute, is also elaborated. It is also shown that the problem of testing coprimality of two 2 × 2 matrices is equvialent to testing coprimality of a pair of triangular matrices, which can be done almost by inspection. Also considered is the case of 3 × 3 triangular matrices and their adjugates, which have potential applications in three dimensional signal processing.