Quadratic Realizability of Palindromic Matrix Polynomials

De Teran, Fernando and Dopico, Froilan and Mackey, D. Steven and Perovic, Vasilije (2017) Quadratic Realizability of Palindromic Matrix Polynomials. [MIMS Preprint]

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Abstract

Let $\cL = (\cL_1,\cL_2)$ be a list consisting of a sublist $\cL_1$ of powers of irreducible (monic) scalar polynomials over an algebraically closed field $\FF$, and a sublist $\cL_2$ of nonnegative integers. For an arbitrary such list $\cL$, we give easily verifiable necessary and sufficient conditions for $\cL$ to be the list of elementary divisors and minimal indices of some $T$-palindromic quadratic matrix polynomial with entries in the field $\FF$. For $\cL$ satisfying these conditions, we show how to explicitly construct a $T$-palindromic quadratic matrix polynomial having $\cL$ as its structural data; that is, we provide a $T$-palindromic quadratic realization of $\cL$. Our construction of $T$-palindromic realizations is accomplished by taking a direct sum of low bandwidth $T$-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of $T$-palindromic quadratic polynomials is that all even grade $T$-palindromic matrix polynomials have a $T$-palindromic strong quadratification. Finally, using a particular M\"{o}bius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with $T$-even structure.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix polynomials, quadratic realizability, elementary divisors, minimal indices, quasi-canonical form, quadratifications, $T$-palindromic, inverse problem
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: 08 Oct 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2584

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