On different concepts for the linearization of matrix polynomials and canonical decompositions of structured matrices with respect to indefinite sesquilinear forms

Philip Saltenberger

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Philip Saltenberger, On different concepts for the linearization of matrix polynomials and canonical decompositions of structured matrices with respect to indefinite sesquilinear forms (2019), Logos Verlag, Berlin, ISBN: 9783832588922

Beschreibung / Abstract

In this thesis, a novel framework for the construction and analysis of strong linearizations for matrix polynomials is presented. Strong linearizations provide the standard means to transform polynomial eigenvalue problems into equivalent generalized eigenvalue problems while preserving the complete finite and infinite eigenstructure of the problem. After the transformation, the QZ algorithm or special methods appropriate for structured linearizations can be applied for finding the eigenvalues efficiently.

The block Kronecker ansatz spaces proposed here establish an innovative and flexible approach for the construction of strong linearizations in the class of strong block minimal bases pencils. Moreover, they represent a new vector-space-setting for linearizations of matrix polynomials that additionally provides a common basis for various existing techniques on this task (such as Fiedler-linearizations). New insights on their relations, similarities and differences are revealed. The generalized eigenvalue problems obtained often allow for an efficient numerical solution. This is discussed with special attention to structured polynomial eigenvalue problems whose linearizations are structured as well.

Structured generalized eigenvalue problems may also lead to equivalent structured (standard) eigenvalue problems. Thereby, the transformation produces matrices that can often be regarded as selfadjoint or skewadjoint with respect to some indefinite inner product. Based on this observation, normal matrices in indefinite inner product spaces and their spectral properties are studied and analyzed. Multiplicative and additive canonical decompositions respecting the matrix structure induced by the inner product are established.

Inhaltsverzeichnis

  • BEGINN
  • Zusammenfassung
  • List of Symbols
  • Introduction
  • Publications
  • Acknowledgement
  • A primer on matrix polynomials
  • Matrices, matrix polynomials and polynomial matrices
  • Eigenvalues and eigenvectors of matrix polynomials
  • Linearizations of matrix polynomials
  • Linearizations from vector spaces and Fiedler-pencils
  • Outlook on Chapters 3 to 7
  • Block Kronecker ansatz spaces
  • Introduction of G+1(A) and basic properties
  • Advanced structural analysis of G+1(A)
  • The recovery of eigenvectors
  • Block Kronecker pencils and Fiedler-like linearizations
  • Double block Kronecker ansatz spaces
  • Definition of DG+1(A) and basic properties
  • Advanced structural analysis of DG+1(A)
  • The superpartition principle
  • The inclusion relation
  • The block-symmetric ansatz spaces BG+ 1(A)
  • A note on symmetric linearizations for symmetric matrix polynomials
  • Conclusions
  • Solving generalized eigenproblems
  • Shift-and-invert reformulation
  • Solving linear systems L()x = z
  • Structured polynomial eigenproblems
  • Algorithms for symmetric GEPs
  • The connection between L1(A), L2(A) and G+1(A)
  • Block Kronecker ansatz spaces and the spaces L1(A) and L2(A)
  • The central relation between L1(A), L2(A) and G+1(A)
  • Conclusions
  • Generalized ansatz spaces for orthogonal bases
  • The basic framework
  • The recovery of eigen- and nullspaces
  • A note on singular matrix polynomials
  • Double generalized ansatz spaces
  • The eigenvector exclusion theorem
  • Conclusions
  • Sesquilinear forms
  • Introduction and basic definitions
  • Special classes of B-normal matrices
  • Forms from generalized permutation matrices
  • Outlook on Chapters 9 and 10
  • Automorphic diagonalization of B-normal matrices
  • Preliminary results on common eigenspaces
  • An eigenbasis ordering
  • Towards an automorphic diagonalization
  • Frequently arising indefinite forms: examples
  • Unitary structure-preserving diagonalization
  • Conclusions
  • Structured normal matrices
  • Lagrangian subspaces
  • Normal perhermitian matrices
  • Normal Hamiltonian matrices
  • Conclusions
  • Bibliography
  • Index

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