Tisseur, Francoise and Garvey, Seamus D. and Munro, Christopher
(2009)
*Deflating Quadratic Matrix Polynomials with Structure Preserving Transformations.*
[MIMS Preprint]

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## Abstract

Given a pair of distinct \e s $(\l_1,\l_2)$ of an $\nbyn$ quadratic matrix polynomial $Q(\l)$ with nonsingular leading coefficient and their corresponding \ev s, we show how to transform $Q(\l)$ into a quadratic of the form $\twobytwoa{\Qd(\l)}{0}{0}{q(\l)}$ having the same \e s as $Q(\l)$, with $\Qd(\l)$ an $(n-1)\times (n-1)$ quadratic matrix \py\ and $q(\l)$ a scalar quadratic \py\ with roots $\l_1$ and $\l_2$. This block diagonalization cannot be achieved by a similarity transformation applied directly to $Q(\l)$ unless the \ev s corresponding to $\l_1$ and $\l_2$ are parallel. We identify conditions under which we can construct a family of $2n\times 2n$ elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of $Q(\l)$, (c) preserve the block structure of a large class of block symmetric linearizations of $Q(\l)$, thereby defining new quadratic matrix polynomials $Q_1(\l)$ that have the same \e s as $Q(\l)$, (d) yield quadratics $Q_1(\l)$ with the property that their \ev s associated with $\l_1$ and $\l_2$ are parallel and hence can subsequently be deflated by a similarity applied directly to $Q_1(\l)$. This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | quadratic eigenvalue problem, linearization, structure preserving transformation, deflation |

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 Françoise Tisseur |

Date Deposited: | 10 Mar 2009 |

Last Modified: | 08 Nov 2017 18:18 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/1242 |

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