Analysis of Structured Polynomial Eigenvalue Problems

  • Maha Al-Ammari

Student thesis: Phd

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 classification of Hermitian matrixpolynomials whose eigenvalues belong to the extended real line, witheach eigenvalue being of definite type, is provided first. We call such polynomials quasidefinite. Definite pencils, definitizable pencils, overdamped quadratics, gyroscopically stabilized quadratics, (quasi)hyperbolic and definite matrix polynomials are all quasidefinite. 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 matrixpolynomial 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 field 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 [NLEVP, 2010] by generatingquadratic and cubic quasidefinite matrix polynomials in differentsubclasses from some given spectral data by solving an appropriateinverse eigenvalue problem. For the quadratic case, we employ available algorithms to provide tridiagonal definite matrix polynmials.
Date of Award31 Dec 2011
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorFrancoise Tisseur (Supervisor) & Catherine Powell (Supervisor)

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