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.
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 |