Fiedler Companion Linearizations and the Recovery of Minimal Indices

De Teran, Fernando and Dopico, Froilan M. and Mackey, D. Steven (2009) Fiedler Companion Linearizations and the Recovery of Minimal Indices. [MIMS Preprint]

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Abstract

A standard way of dealing with a matrix polynomial $P(\lambda)$ is to convert it into an equivalent matrix pencil -- a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.

Item Type: MIMS Preprint
Uncontrolled Keywords: singular matrix polynomials, matrix pencils, minimal indices, minimal bases, linearization, recovery of eigenvectors, Fiedler pencils, companion forms
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: 31 Mar 2010
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1428

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