Skew-symmetric matrix polynomials and their Smith forms

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Abstract

Two canonical forms for skew-symmetric matrix polynomials over arbitrary fields are characterized—the Smith form, and its skew-symmetric variant obtained via unimodular congruences. Applications include the analysis of the eigenvalue and elementary divisor structure of products of two skew-symmetric matrices, the derivation of a Smith-McMillan-like canonical form for skew-symmetric rational matrices, and the construction of minimal symmetric factorizations of skew-symmetric rational matrices. A sufficient condition for the existence of solutions to matrix polynomial Sylvester equations, and results on the existence and construction of structured linearizations for regular and singular skew-symmetric matrix polynomials are also presented.

AMS classification

65F15
15A18
15A21
15A54
15A57

Keywords

Smith form
Skew-symmetric matrix polynomial
Structured linearization
Unimodular congruence
Smith-McMillan form
Minimal symmetric factorization

Cited by (0)

1

Supported by National Science Foundation grant DMS-1016224. Support from Deutsche Forschungsgemeinschaft through DFG Research Center Matheon during research visits to TU Berlin is gratefully acknowledged.

2

Supported by Deutsche, Forschungsgemeinschaft, through DFG Research Center Matheon, ’Mathematics for key technologies’ in Berlin.