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Handbook of Linear Algebra





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ISBN 9781138199897
Published June 28, 2017 by Chapman and Hall/CRC
1904 Pages - 145 B/W Illustrations

 
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Book Description

With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.

New to the Second Edition

  • Separate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of matrices, generalized inverses, matrices over finite fields, invariant subspaces, representations of quivers, and spectral sets
  • New chapters on combinatorial matrix theory topics, such as tournaments, the minimum rank problem, and spectral graph theory, as well as numerical linear algebra topics, including algorithms for structured matrix computations, stability of structured matrix computations, and nonlinear eigenvalue problems
  • More chapters on applications of linear algebra, including epidemiology and quantum error correction
  • New chapter on using the free and open source software system Sage for linear algebra
  • Additional sections in the chapters on sign pattern matrices and applications to geometry
  • Conjectures and open problems in most chapters on advanced topics

Highly praised as a valuable resource for anyone who uses linear algebra, the first edition covered virtually all aspects of linear algebra and its applications. This edition continues to encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, and applications of linear algebra to various disciplines while also covering up-to-date software packages for linear algebra computations.

Table of Contents

Linear Algebra
Linear Algebra
Vectors, Matrices, and Systems of Linear Equations Jane Day
Linear Independence, Span, and Bases Mark Mills
Linear Transformations Francesco Barioli
Determinants and Eigenvalues Luz M. DeAlba
Inner Product Spaces, Orthogonal Projection, Least Squares, and Singular Value Decomposition Lixing Han and Michael Neumann
Canonical Forms Leslie Hogben
Other Canonical Forms Roger A. Horn and Vladimir V. Sergeichuk
Unitary Similarity, Normal Matrices, and Spectral Theory Helene Shapiro
Hermitian and Positive Definite Matrices Wayne Barrett
Nonnegative and Stochastic Matrices Uriel G. Rothblum
Partitioned Matrices Robert Reams

Topics in Linear Algebra
Schur Complements Roger A. Horn and Fuzhen Zhang
Quadratic, Bilinear, and Sesquilinear Forms Raphael Loewy
Multilinear Algebra J.A. Dias da Silva and Armando Machado
Tensors and Hypermatrices Lek-Heng Lim
Matrix Equalities and Inequalities Michael Tsatsomeros
Functions of Matrices Nicholas J. Higham
Matrix Polynomials Jorg Liesen and Christian Mehl
Matrix Equations Beatrice Meini
Invariant Subspaces G.W. Stewart
Matrix Perturbation Theory Ren-Cang Li
Special Types of Matrices Albrecht Bottcher and Ilya Spitkovsky
Pseudospectra Mark Embree
Singular Values and Singular Value Inequalities Roy Mathias
Numerical Range Chi-Kwong Li
Matrix Stability and Inertia Daniel Hershkowitz
Generalized Inverses of Matrices Yimin Wei
Inverse Eigenvalue Problems Alberto Borobia
Totally Positive and Totally Nonnegative Matrices Shaun M. Fallat
Linear Preserver Problems Peter Semrl
Matrices over Finite Fields J. D. Botha
Matrices over Integral Domains Shmuel Friedland
Similarity of Families of Matrices Shmuel Friedland
Representations of Quivers and Mixed Graphs Roger A. Horn and Vladimir V. Sergeichuk
Max-Plus Algebra Marianne Akian, Ravindra Bapat, and Stephane Gaubert
Matrices Leaving a Cone Invariant Bit-Shun Tam and Hans Schneider
Spectral Sets Catalin Badea and Bernhard Beckermann

Combinatorial Matrix Theory and Graphs
Combinatorial Matrix Theory
Combinatorial Matrix Theory Richard A. Brualdi
Matrices and Graphs Willem H. Haemers
Digraphs and Matrices Jeffrey L. Stuart
Bipartite Graphs and Matrices Bryan L. Shader
Sign Pattern Matrices Frank J. Hall and Zhongshan Li

Topics in Combinatorial Matrix Theory
Permanents Ian M. Wanless
D-Optimal Matrices Michael G. Neubauer and William Watkins
Tournaments T.S. Michael
Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs Shaun M. Fallat and Leslie Hogben
Spectral Graph Theory Steve Butler and Fan Chung
Algebraic Connectivity Steve Kirkland
Matrix Completion Problems Luz M. DeAlba, Leslie Hogben, and Amy Wangsness Wehe

Numerical Methods
Numerical Methods for Linear Systems
Vector and Matrix Norms, Error Analysis, Efficiency, and Stability Ralph Byers and Biswa Nath Datta
Matrix Factorizations and Direct Solution of Linear Systems Christopher Beattie
Least Squares Solution of Linear Systems Per Christian Hansen and Hans Bruun Nielsen
Sparse Matrix Methods Esmond G. Ng
Iterative Solution Methods for Linear Systems Anne Greenbaum

Numerical Methods for Eigenvalues
Symmetric Matrix Eigenvalue Techniques Ivan Slapnicar
Unsymmetric Matrix Eigenvalue Techniques David S. Watkins
The Implicitly Restarted Arnoldi Method D.C. Sorensen
Computation of the Singular Value Decomposition Alan Kaylor Cline and Inderjit S. Dhillon
Computing Eigenvalues and Singular Values to High Relative Accuracy Zlatko Drmac
Nonlinear Eigenvalue Problems Heinrich Voss

Topics in Numerical Linear Algebra
Fast Matrix Multiplication Dario A. Bini
Fast Algorithms for Structured Matrix Computations Michael Stewart
Structured Eigenvalue Problems | Structure-Preserving Algorithms, Structured Error Analysis Heike Fassbender
Large-Scale Matrix Computations Roland W. Freund

Linear Algebra in Other Disciplines
Applications to Physical and Biological Sciences
Linear Algebra and Mathematical Physics Lorenzo Sadun
Linear Algebra in Biomolecular Modeling Zhijun Wu
Linear Algebra in Mathematical Population Biology and Epidemiology Fred Brauer and Carlos Castillo-Chavez

Applications to Optimization
Linear Programming Leonid N. Vaserstein
Semidefinite Programming Henry Wolkowicz

Applications to Probability and Statistics
Random Vectors and Linear Statistical Models Simo Puntanen and George P.H. Styan
Multivariate Statistical Analysis Simo Puntanen, George A.F. Seber, and George P.H. Styan
Markov Chains Beatrice Meini

Applications to Computer Science
Coding Theory Joachim Rosenthal and Paul Weiner
Quantum Computation Zijian Diao
Operator Quantum Error Correction Chi-Kwong Li, Yiu-Tung Poon, and Nung-Sing Sze
Information Retrieval and Web Search Amy N. Langville and Carl D. Meyer
Signal Processing Michael Stewart

Applications to Analysis
Differential Equations and Stability Volker Mehrmann and Tatjana Stykel
Dynamical Systems and Linear Algebra Fritz Colonius and Wolfgang Kliemann
Control Theory Peter Benner
Fourier Analysis Kenneth Howell

Applications to Geometry
Geometry Mark Hunacek
Some Applications of Matrices and Graphs in Euclidean Geometry Miroslav Fiedler

Applications to Algebra
Matrix Groups Peter J. Cameron
Group Representations Randall R. Holmes and Tin-Yau Tam
Nonassociative Algebras Murray R. Bremner, Lucia I. Murakami, and Ivan P. Shestakov
Lie Algebras Robert Wilson

Computational Software
Interactive Software for Linear Algebra
MATLAB Steven J. Leon
Linear Algebra in Maple David J. Jeffrey and Robert M. Corless
Mathematica Heikki Ruskeep’a’a
Sage Robert A. Beezer, Robert Bradshaw, Jason Grout, and William Stein

Packages of Subroutines for Linear Algebra
BLAS Jack Dongarra, Victor Eijkhout, and Julien Langou
LAPACK Zhaojun Bai, James Demmel, Jack Dongarra, Julien Langou, and Jenny Wang
Use of ARPACK and EIGS D.C. Sorensen
Summary of Software for Linear Algebra Freely Available on the Web Jack Dongarra, Victor Eijkhout, and Julien Langou

Glossary

Notation Index

Index

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Editor(s)

Biography

Leslie Hogben is the Dio Lewis Holl Chair in Applied Mathematics and Professor of Mathematics at Iowa State University (ISU), and the Associate Director for Diversity of the American Institute of Mathematics. She received her B.A. from Swarthmore College in 1974 and her Ph. D. in 1978 from Yale University under the direction of Nathan Jacobson. Although originally working in nonassociative algebras, in the mid-1990s, she changed her focus to linear algebra.
Dr. Hogben is the author of more than 60 research papers and particularly enjoys introducing students to mathematical research. She has or is advising three postdoctoral associates, 11 doctoral students, 12 master's students, and 30 undergraduate researchers. She is the co-director of the NSF-sponsored ISU Math REU and developed an early graduate research course for mathematics and applied mathematics graduate students at ISU.
Dr. Hogben is a frequent co-organizer of meetings, workshops, and special sessions/mini-symposia. She is the Secretary/Treasurer of the International Linear Algebra Society and an associate editor of the journals Linear Algebra and its Applications and Electronic Journal of Linear Algebra.

Reviews

"The second edition is substantially expanded … A huge amount of thought has gone into the book. … Leslie Hogben has done an outstanding job to produce a book of this size in a uniform style with such a high standard of editing and typesetting. … this is certainly a book to ask your library to order and maybe even to purchase yourself."
—Nick Higham (University of Manchester) on his blog http://nickhigham.wordpress.com, March 2014

Praise for the First Edition:
"… a valuable compendium of information on virtually all aspects of linear algebra and its applications. …This is a Herculean labor of love on the editor’s part, a successful effort that should be appreciated and applauded by anyone working and/or teaching in this important area of mathematics. … Every library that supports mathematics and science departments should have this encyclopedic work on its shelves."
—Henry Ricardo, MAA Reviews, June 2007

"Without doubts, it will be of great help for everybody who needs linear algebra in his profession, or for instructors or students looking for concepts, results, or examples."
—H. Mitsch, Monatshefte fur Math, 2007

"This handbook covers all the major topics of linear algebra at both graduate and undergraduate level, as well as their applications and related software packages. The handbook will be without a doubt a valuable resource for anyone using linear algebra, i.e. basically for any mathematician."
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