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Reducing Matrix Polynomials to Simpler Forms Zaballa, Ion
Description
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 will illustrate how to use these reduced forms to solve parameterized linear systems.
Item Metadata
Title |
Reducing Matrix Polynomials to Simpler Forms
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-07-08T10:00
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Description |
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 will illustrate how to use these reduced forms
to solve parameterized linear systems.
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Extent |
29 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Euskal Herriko Unibertsitatea
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Series | |
Date Available |
2018-01-04
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0362587
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International