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Skew-Symmetric Matrix Polynomials and their Smith Forms

Please always quote using this URN:urn:nbn:de:0296-matheon-11445
  • We characterize the Smith form of skew-symmetric matrix polynomials over an arbitrary field $\F$, showing that all elementary divisors occur with even multiplicity. Restricting the class of equivalence transformations to unimodular congruences, a Smith-like skew-symmetric canonical form for skew-symmetric matrix polynomials is also obtained. These results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew-symmetric matrices, as well as the existence of structured linearizations for skew-symmetric matrix polynomials. By contrast with other classes of structured matrix polynomials (e.g., alternating or palindromic polynomials), every regular skew-symmetric matrix polynomial is shown to have a structured strong linearization. While there are singular skew-symmetric polynomials of even degree for which a structured linearization is impossible, for each odd degree we develop a skew-symmetric companion form that uniformly provides a structured linearization for every regular and singular skew-symmetric polynomial of that degree. Finally, the results are applied to the construction of minimal symmetric factorizations of skew-symmetric rational matrices.

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Metadaten
Author:Christian Mehl, Volker Mehrmann, D. Steven Mackey, Niloufer Mackey
URN:urn:nbn:de:0296-matheon-11445
Referee:Fredi Tröltzsch
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/07/02
Release Date:2012/07/02
Tag:Smith form; matrix polynomial; skew-symmetric
Institute:Technische Universität Berlin
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F15 Eigenvalues, eigenvectors
Preprint Number:967
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