Factorizations fornD polynomial matrices

In this paper, a constructive general matrix factorization scheme is developed for extracting a nontrivial factor from a givennD (n>2) polynomial matrix whose maximal order minors satisfy certain conditions. It is shown that three classes ofnD polynomial matrices admit this kind of general matrix factorization. It turns out that minor prime factorization and determinantal factorization are two interesting special cases of the proposal general factorization. As a consequence, the paper provides a partial solution to an open problem of minor prime factorization as well as to a recent conjecture on minor prime factorizability fornD polynomial matrices. Three illustrative examples are worked out in detail.

[1]  Virendra Sule,et al.  Feedback Stabilization Over Commutative Rings: The Matrix Case , 1994 .

[2]  A. C. Tan,et al.  On a generalized factorization problem for structurally passive synthesis of digital filters , 1989 .

[3]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[4]  H. Park,et al.  An Algorithmic Proof of Suslin′s Stability Theorem for Polynomial Rings , 1994, alg-geom/9405003.

[5]  Dante C. Youla,et al.  Notes on n-Dimensional System Theory , 1979 .

[6]  B. Sturmfels,et al.  Algorithms for the Quillen-Suslin theorem , 1992 .

[7]  N. Bose.  Applied multidimensional systems theory , 1982 .

[8]  Martin Vetterli,et al.  Gröbner Bases and Multidimensional FIR Multirate Systems , 1997, Multidimens. Syst. Signal Process..

[9]  Sun-Yuan Kung,et al.  New results in 2-D systems theory, part I: 2-D polynomial matrices, factorization, and coprimeness , 1977, Proceedings of the IEEE.

[10]  N. K. Bose,et al.  Multidimensional FIR filter bank design using Grobner bases , 1999 .

[11]  Zhiping Lin,et al.  Feedback Stabilizability of MIMO n-D Linear Systems , 1998, Multidimens. Syst. Signal Process..

[12]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[13]  Ettore Fornasini,et al.  nD Polynomial Matrices with Applications to Multidimensional Signal Analysis , 1997, Multidimens. Syst. Signal Process..

[14]  D. Quillen.  Projective modules over polynomial rings , 1976 .

[15]  Rajnikant V. Patel,et al.  Computation of simple and group factors of multivariate polynomials , 1997 .

[16]  Thomas Kailath,et al.  Linear Systems , 1980 .

[17]  Zhiping Lin.  On primitive factorizations for 3-D polynomial matrices , 1992 .

[18]  Zhiping Lin,et al.  Feedback stabilization of MIMO 3-D linear systems , 1999, IEEE Trans. Autom. Control..

[19]  J. Guiver,et al.  Polynomial matrix primitive factorization over arbitrary coefficient field and related results , 1982 .

[20]  Paul S. Wang,et al.  Factoring multivariate polynomials over the integers , 1973, SIGS.

[21]  William H. Press,et al.  Numerical recipes in C , 2002 .

[22]  Dante C. Youla,et al.  The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices , 1984 .

[23]  Zhiping Lin.  On matrix fraction descriptions of multivariable linear n-D systems , 1988 .

[24]  Zhiping Lin,et al.  On primitive factorizations for n-D polynomial matrices , 1993, 1993 IEEE International Symposium on Circuits and Systems.

[25]  U. Oberst,et al.  Transfer Operators and State Spaces for Discrete Multidimensional Linear Systems , 1999 .

[26]  Nikos E. Mastorakis,et al.  A general factorization method for multivariable polynomials , 1994, Multidimens. Syst. Signal Process..

[27]  Zhiping Lin,et al.  Notes on n-D Polynomial Matrix Factorizations , 1999, Multidimens. Syst. Signal Process..