Functions of Matrices

Functions of Matrices

Author: Nicholas J. Higham

Publisher: SIAM

Published: 2008-01-01

Total Pages: 445

ISBN-13: 0898717779

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A thorough and elegant treatment of the theory of matrix functions and numerical methods for computing them, including an overview of applications, new and unpublished research results, and improved algorithms. Key features include a detailed treatment of the matrix sign function and matrix roots; a development of the theory of conditioning and properties of the Fre;chet derivative; Schur decomposition; block Parlett recurrence; a thorough analysis of the accuracy, stability, and computational cost of numerical methods; general results on convergence and stability of matrix iterations; and a chapter devoted to the f(A)b problem. Ideal for advanced courses and for self-study, its broad content, references and appendix also make this book a convenient general reference. Contains an extensive collection of problems with solutions and MATLAB implementations of key algorithms.


Monotone Matrix Functions and Analytic Continuation

Monotone Matrix Functions and Analytic Continuation

Author: W.F.Jr. Donoghue

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 191

ISBN-13: 3642657559

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A Pick function is a function that is analytic in the upper half-plane with positive imaginary part. In the first part of this book we try to give a readable account of this class of functions as well as one of the standard proofs of the spectral theorem based on properties of this class. In the remainder of the book we treat a closely related topic: Loewner's theory of monotone matrix functions and his analytic continuation theorem which guarantees that a real function on an interval of the real axis which is a monotone matrix function of arbitrarily high order is the restriction to that interval of a Pick function. In recent years this theorem has been complemented by the Loewner-FitzGerald theorem, giving necessary and sufficient conditions that the continuation provided by Loewner's theorem be univalent. In order that our presentation should be as complete and trans parent as possible, we have adjoined short chapters containing the in formation about reproducing kernels, almost positive matrices and certain classes of conformal mappings needed for our proofs.


Loewner's Theorem on Monotone Matrix Functions

Loewner's Theorem on Monotone Matrix Functions

Author: Barry Simon

Publisher: Springer Nature

Published: 2019-08-29

Total Pages: 445

ISBN-13: 3030224228

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This book provides an in depth discussion of Loewner’s theorem on the characterization of matrix monotone functions. The author refers to the book as a ‘love poem,’ one that highlights a unique mix of algebra and analysis and touches on numerous methods and results. The book details many different topics from analysis, operator theory and algebra, such as divided differences, convexity, positive definiteness, integral representations of function classes, Pick interpolation, rational approximation, orthogonal polynomials, continued fractions, and more. Most applications of Loewner’s theorem involve the easy half of the theorem. A great number of interesting techniques in analysis are the bases for a proof of the hard half. Centered on one theorem, eleven proofs are discussed, both for the study of their own approach to the proof and as a starting point for discussing a variety of tools in analysis. Historical background and inclusion of pictures of some of the main figures who have developed the subject, adds another depth of perspective. The presentation is suitable for detailed study, for quick review or reference to the various methods that are presented. The book is also suitable for independent study. The volume will be of interest to research mathematicians, physicists, and graduate students working in matrix theory and approximation, as well as to analysts and mathematical physicists.


Matrix Functions And Matrix Equations

Matrix Functions And Matrix Equations

Author: Zhaojun Bai

Publisher: World Scientific

Published: 2015-09-04

Total Pages: 146

ISBN-13: 9814675784

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Matrix functions and matrix equations are widely used in science, engineering and social sciences due to the succinct and insightful way in which they allow problems to be formulated and solutions to be expressed. This book covers materials relevant to advanced undergraduate and graduate courses in numerical linear algebra and scientific computing. It is also well-suited for self-study. The broad content makes it convenient as a general reference to the subjects.


Matrices, Moments and Quadrature with Applications

Matrices, Moments and Quadrature with Applications

Author: Gene H. Golub

Publisher: Princeton University Press

Published: 2009-12-07

Total Pages: 376

ISBN-13: 1400833884

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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.


Matrix Theory

Matrix Theory

Author: Fuzhen Zhang

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 290

ISBN-13: 1475757972

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This volume concisely presents fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. For many theorems several different proofs are given. The only prerequisites are a decent background in elementary linear algebra and calculus.


Model Order Reduction: Theory, Research Aspects and Applications

Model Order Reduction: Theory, Research Aspects and Applications

Author: Wilhelmus H. Schilders

Publisher: Springer Science & Business Media

Published: 2008-08-27

Total Pages: 471

ISBN-13: 3540788417

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The idea for this book originated during the workshop “Model order reduction, coupled problems and optimization” held at the Lorentz Center in Leiden from S- tember 19–23, 2005. During one of the discussion sessions, it became clear that a book describing the state of the art in model order reduction, starting from the very basics and containing an overview of all relevant techniques, would be of great use for students, young researchers starting in the ?eld, and experienced researchers. The observation that most of the theory on model order reduction is scattered over many good papers, making it dif?cult to ?nd a good starting point, was supported by most of the participants. Moreover, most of the speakers at the workshop were willing to contribute to the book that is now in front of you. The goal of this book, as de?ned during the discussion sessions at the workshop, is three-fold: ?rst, it should describe the basics of model order reduction. Second, both general and more specialized model order reduction techniques for linear and nonlinear systems should be covered, including the use of several related numerical techniques. Third, the use of model order reduction techniques in practical appli- tions and current research aspects should be discussed. We have organized the book according to these goals. In Part I, the rationale behind model order reduction is explained, and an overview of the most common methods is described.


Introduction to Applied Linear Algebra

Introduction to Applied Linear Algebra

Author: Stephen Boyd

Publisher: Cambridge University Press

Published: 2018-06-07

Total Pages: 477

ISBN-13: 1316518965

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A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.


Handbook of Green's Functions and Matrices

Handbook of Green's Functions and Matrices

Author: V. D. Şeremet

Publisher: Witpress

Published: 2003

Total Pages: 312

ISBN-13:

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Designed for graduate and postgraduate students investigating such areas as elasticity, thermoelasticity, mechanics, heat conduction, elector and magneto conduction, electronics, radio-physics, hydrodynamics, and conduction of moisture, the text will also be of interest to engineers and researchers working in these fields.


Introduction to Matrix Analysis and Applications

Introduction to Matrix Analysis and Applications

Author: Fumio Hiai

Publisher: Springer Science & Business Media

Published: 2014-02-06

Total Pages: 337

ISBN-13: 3319041509

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Matrices can be studied in different ways. They are a linear algebraic structure and have a topological/analytical aspect (for example, the normed space of matrices) and they also carry an order structure that is induced by positive semidefinite matrices. The interplay of these closely related structures is an essential feature of matrix analysis. This book explains these aspects of matrix analysis from a functional analysis point of view. After an introduction to matrices and functional analysis, it covers more advanced topics such as matrix monotone functions, matrix means, majorization and entropies. Several applications to quantum information are also included. Introduction to Matrix Analysis and Applications is appropriate for an advanced graduate course on matrix analysis, particularly aimed at studying quantum information. It can also be used as a reference for researchers in quantum information, statistics, engineering and economics.