Quadratic Realizability of Palindromic Matrix Polynomials: the Real Case

Perovic, Vasilije and Mackey, D. Steven (2021) Quadratic Realizability of Palindromic Matrix Polynomials: the Real Case. [MIMS Preprint]

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Abstract

Let $\cL = (\cL_1,\cL_2)$ be a list consisting of structural data for a matrix polynomial; here $\cL_1$ is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field $\RR$, and $\cL_2$ is a sublist of nonnegative integers. For an arbitrary such $\cL$, we give easy-to-check necessary and sufficient conditions for $\cL$ to be the list of elementary divisors and minimal indices of some real $T$-palindromic quadratic matrix polynomial. For a list $\cL$ satisfying these conditions, we show how to explicitly build a real $T$-palindromic quadratic matrix polynomial having $\cL$ as its structural data; that is, we provide a $T$-palindromic quadratic realization of $\cL$ over $\RR$. A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list $\cL$.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	matrix polynomials, real quadratic realizability, elementary divisors, minimal indices, T-palindromic, inverse problem
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
Depositing User:	Dr. D. Steven Mackey
Date Deposited:	10 Dec 2021 11:51
Last Modified:	10 Dec 2021 11:51
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2840

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