2nd Edition

Handbook of Linear Algebra

Edited By Leslie Hogben Copyright 2014
    1904 Pages 145 B/W Illustrations
    by Chapman & Hall

    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.

    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


    Notation Index



    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.

    "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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