On different concepts for the linearization of matrix polynomials and canonical decompositions of structured matrices with respect to indefinite sesquilinear forms
Title | On different concepts for the linearization of matrix polynomials and canonical decompositions of structured matrices with respect to indefinite sesquilinear forms PDF eBook |
Author | Philip Saltenberger |
Publisher | Logos Verlag Berlin GmbH |
Pages | 191 |
Release | 2019-05-30 |
Genre | Mathematics |
ISBN | 3832549145 |
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.
Matrix Polynomials
Title | Matrix Polynomials PDF eBook |
Author | Israel Gohberg |
Publisher | |
Pages | 440 |
Release | 1982 |
Genre | Mathematics |
ISBN |
This book provides a comprehensive treatment of the theory of matrix polynomials. The theory developed here is a natural extension to polynomials of higher degrees, and forms an important new part of linear algebra for which the main concepts and results have been arrived at during the past five years.
Matrix Theory: A Second Course
Title | Matrix Theory: A Second Course PDF eBook |
Author | James M. Ortega |
Publisher | Springer Science & Business Media |
Pages | 278 |
Release | 1987-02-28 |
Genre | Mathematics |
ISBN | 9780306424335 |
Linear algebra and matrix theory are essentially synonymous terms for an area of mathematics that has become one of the most useful and pervasive tools in a wide range of disciplines. It is also a subject of great mathematical beauty. In consequence of both of these facts, linear algebra has increasingly been brought into lower levels of the curriculum, either in conjunction with the calculus or separate from it but at the same level. A large and still growing number of textbooks has been written to satisfy this need, aimed at students at the junior, sophomore, or even freshman levels. Thus, most students now obtaining a bachelor's degree in the sciences or engineering have had some exposure to linear algebra. But rarely, even when solid courses are taken at the junior or senior levels, do these students have an adequate working knowledge of the subject to be useful in graduate work or in research and development activities in government and industry. In particular, most elementary courses stop at the point of canonical forms, so that while the student may have "seen" the Jordan and other canonical forms, there is usually little appreciation of their usefulness. And there is almost never time in the elementary courses to deal with more specialized topics like nonnegative matrices, inertia theorems, and so on. In consequence, many graduate courses in mathematics, applied mathe matics, or applications develop certain parts of matrix theory as needed.
Structured Matrices and Polynomials
Title | Structured Matrices and Polynomials PDF eBook |
Author | Victor Y. Pan |
Publisher | Springer Science & Business Media |
Pages | 299 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461201292 |
This user-friendly, engaging textbook makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. The book goes beyond research frontiers and, apart from very recent research articles, includes previously unpublished results.
Numerical Methods for General and Structured Eigenvalue Problems
Title | Numerical Methods for General and Structured Eigenvalue Problems PDF eBook |
Author | Daniel Kressner |
Publisher | Springer Science & Business Media |
Pages | 272 |
Release | 2006-01-20 |
Genre | Mathematics |
ISBN | 3540285024 |
This book is about computing eigenvalues, eigenvectors, and invariant subspaces of matrices. Treatment includes generalized and structured eigenvalue problems and all vital aspects of eigenvalue computations. A unique feature is the detailed treatment of structured eigenvalue problems, providing insight on accuracy and efficiency gains to be expected from algorithms that take the structure of a matrix into account.
Matrix Canonical Forms
Title | Matrix Canonical Forms PDF eBook |
Author | S. Gill Williamson |
Publisher | Createspace Independent Publishing Platform |
Pages | 112 |
Release | 2014-07-07 |
Genre | Mathematics |
ISBN | 9781500289508 |
This material is a rewriting of notes handed out by me to beginning graduate students in seminars in combinatorial mathematics (Department of Mathematics, University of California San Diego). Topics covered in this seminar were in algebraic and algorithmic combinatorics. Solid skills in linear and multilinear algebra were required of students in these seminars - especially in algebraic combinatorics. I developed these notes to review the students' undergraduate linear algebra and improve their proof skills. We focused on a careful development of the general matrix canonical forms as a training ground.
Perturbation theory for linear operators
Title | Perturbation theory for linear operators PDF eBook |
Author | Tosio Kato |
Publisher | Springer Science & Business Media |
Pages | 610 |
Release | 2013-06-29 |
Genre | Mathematics |
ISBN | 3662126788 |