## 2012.70: Skew-symmetric matrix polynomials and their Smith forms

2012.70:
D. Steven Mackey, Niloufer Mackey, Christian Mehl and Volker Mehrmann
(2012)
*Skew-symmetric matrix polynomials and their Smith forms.*

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## Abstract

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 |
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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 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2012.70 |

Deposited By: | Dr. D. Steven Mackey |

Deposited On: | 12 July 2012 |

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