Skew-symmetric matrix polynomials and their Smith forms

Mackey, D. Steven and Mackey, Niloufer and Mehl, Christian and Mehrmann, Volker (2012) Skew-symmetric matrix polynomials and their Smith forms. [MIMS Preprint]

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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.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix polynomial, matrix pencil, compound matrix, Smith form, elementary divisors, invariant polynomials, Jordan structure, skew-symmetric matrix polynomial, structured linearization, companion form, unimodular congruence, skew-symmetric canonical form, Smith-McMillan form, skew-symmetric rational matrix, minimal symmetric factorization.
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr. D. Steven Mackey
Date Deposited: 12 Jul 2012
Last Modified: 08 Nov 2017 18:18

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