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Published April 2011 | public
Journal Article

Coprimality of Certain Families of Integer Matrices

Abstract

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.

Additional Information

© 2010 IEEE. Manuscript received June 24, 2010; revised November 03, 2010; accepted December 21, 2010. Date of publication December 30, 2010; date of current version March 09, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Soontorn Oraintara. This work was supported in part by the ONR grant N00014-08-1-0709, and the California Institute of Technology. The authors would like to thank the reviewers for their many useful suggestions.

Additional details

Created:
August 22, 2023
Modified:
October 23, 2023