Reduction of matrix polynomials to simpler forms

Nakatsukasa, Yuji and Taslaman, Leo and Tisseur, Francoise and Zaballa, Ion (2017) Reduction of matrix polynomials to simpler forms. [MIMS Preprint]

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Abstract

A square matrix can be reduced to simpler form via similarity transformations. Here ``simpler form'' may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Similar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, unimodular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an applications, we illustrate how to use these reduced forms to solve parameterized linear systems.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Fran├žoise Tisseur
Date Deposited: 11 Apr 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2539

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