Al-Ammari, Maha
(2011)
*Analysis of Structured Polynomial Eigenvalue Problems.*
Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.

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## Abstract

This thesis considers Hermitian/symmetric, alternating and palindromic matrix polynomials which all arise frequently in a variety of applications, such as vibration analysis of dynamical systems and optimal control problems. A classication of Hermitian matrix polynomials whose eigenvalues belong to the extended real line, with each eigenvalue being of denite type, is provided rst. We call such polynomials quasidenite. Denite pencils, denitizable pencils, overdamped quadratics, gyroscopically stabilized quadratics, (quasi)hyperbolic and denite matrix polynomials are all quasidenite. We show, using homogeneous rotations, special Hermitian linearizations and a new characterization of hyperbolic matrix polynomials, that the main common thread between these many subclasses is the distribution of their eigenvalue types. We also identify, amongst all quasihyperbolic matrix polynomials, those that can be diagonalized by a congruence transformation applied to a Hermitian linearization of the matrix polynomial while maintaining the structure of the linearization. Secondly, we generalize the notion of self-adjoint standard triples associated with Hermitian matrix polynomials in Gohberg, Lancaster and Rodman's theory of matrix polynomials to present spectral decompositions of structured matrix polynomials in terms of standard pairs (X,T), which are either real or complex, plus a parameter matrix S that acquires particular properties depending on the structure under investigation. These decompositions are mainly an extension of the Jordan canonical form for a matrix over the real or complex eld so we investigate the important special case of structured Jordan triples. Finally, we use the concept of structured Jordan triples to solve a structured inverse polynomial eigenvalue problem. As a consequence, we can enlarge the collection of nonlinear eigenvalue problems by generating quadratic and cubic quasidenite matrix polynomials in dierent subclasses from some given spectral data by solving an appropriate inverse eigenvalue problem. For the quadratic case, we employ available algorithms to provide tridiagonal denite matrix polynomials.

Item Type: | Thesis (Doctoral) |
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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: | 19 Dec 2011 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/1694 |

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