TY - GEN A1 - Mehl, Christian A1 - Mehrmann, Volker A1 - Mackey, D. Steven A1 - Mackey, Niloufer T1 - Skew-Symmetric Matrix Polynomials and their Smith Forms N2 - 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. KW - matrix polynomial KW - skew-symmetric KW - Smith form Y1 - 2012 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1144 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-11445 ER -