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

Vaidyanathan, P. P. and Pal, Piya (2011) A General Approach to Coprime Pairs of Matrices, Based on Minors. IEEE Transactions on Signal Processing, 59 (8). pp. 3536-3548. ISSN 1053-587X.

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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.

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Vaidyanathan, P. P.0000-0003-3003-7042
Additional Information:© 2011 IEEE. Manuscript received December 21, 2010; revised March 19, 2011; accepted April 30, 2011. Date of publication May 10, 2011; date of current version July 13, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jean-Christophe Pesquet. This work was supported in part by the ONR by Grant N00014-08-1-0709 and by the California Institute of Technology.
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Office of Naval Research (ONR)N00014-08-1-0709
Subject Keywords:Adjugate pairs; circulants; commuting matrices; coprime arrays; coprime matrices; minors
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INSPEC Accession Number12108524
Issue or Number:8
Record Number:CaltechAUTHORS:20110829-111302430
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Official Citation:Vaidyanathan, P.P.; Pal, P.; , "A General Approach to Coprime Pairs of Matrices, Based on Minors," Signal Processing, IEEE Transactions on , vol.59, no.8, pp.3536-3548, Aug. 2011 doi: 10.1109/TSP.2011.2152401 URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:25147
Deposited By: Tony Diaz
Deposited On:30 Aug 2011 15:29
Last Modified:09 Mar 2020 13:19

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