Some Properties of Laurent Polynomial Matrices

Sylvie Icart 1 Pierre Comon 2
2 GIPSA-CICS - CICS
GIPSA-DIS - Département Images et Signal
Abstract : In the context of multivariate signal processing, factorizations involving so-called para-unitary matrices are relevant as well demonstrated in the book of Vaidyanathan, and more recently in a series of papers by McWhirter and co-authors. However, known factorizations of matrix polynomials, such as the Smith form, involve unimodular matrices but usual factorizations such as QR, eigenvalue or singular value decompositions, have not been proved to exist for polynomial matrices, if defined with para-unitary matrices, except for very restrictive matrices. It is clear that Cholesky factorization requires square roots, and that EVD and SVD require roots of higher degree polynomials. But one can ask oneself whether the closure of the field of polynomial coefficients is enough or not. It turns out that it is not. Nevertheless, density arguments allow to approximate any polynomial matrix by an SVD-type factorization involving paraunitary polynomial matrices. With that goal, we define the appropriate framework for Laurent polynomial matrices, that is, polynomial matrices with both positive and negative powers in a single variable, particularly the notion of ordrer and degree. We introduce a Smith form for these matrices involving ''L-unimodular'' matrices which are matrices with a monomial non-zero determinant. The 'Elementary Polynomial Givens Rotations' of Foster and McWhirter are of that kind.
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Communication dans un congrès
9th IMA International Conference on Mathematics in Signal Processing, Dec 2012, Birmingham, United Kingdom. pp.4, 2012
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Soumis le : mardi 30 octobre 2012 - 18:08:54
Dernière modification le : mardi 17 mai 2016 - 19:37:57
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Sylvie Icart, Pierre Comon. Some Properties of Laurent Polynomial Matrices. 9th IMA International Conference on Mathematics in Signal Processing, Dec 2012, Birmingham, United Kingdom. pp.4, 2012. <hal-00747253>

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