Smith Forms of Palindromic Matrix Polynomials

Mackey, D. Steven and Mackey, Niloufer and Mehl, Christian and Mehrmann, Volker (2010) Smith Forms of Palindromic Matrix Polynomials. [MIMS Preprint]

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Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular $T$-palindromic matrix polynomials over arbitrary fields. The invariant polynomials are shown to inherit palindromicity, and their structure is described in detail. Jordan structures of palindromic matrix polynomials are characterized, and necessary conditions for the existence of structured linearizations established. In the odd degree case, a constructive procedure for building palindromic linearizations shows that the necessary conditions are sufficient as well. The Smith form for $*$-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of characteristic two are presented.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix polynomial, matrix pencil, structured linearization, Smith form, elementary divisors, invariant polynomials, Jordan structure, palindromic matrix polynomial, compound matrix
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: 13 May 2010
Last Modified: 08 Nov 2017 18:18

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