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Publisher:

Chapman&Hall/CRC

Publication Date:

2007

Number of Pages:

1400

Format:

Hardcover

Series:

Discrete Mathematics and Its Applications 39

Price:

119.95

ISBN:

1584885106

Category:

Handbook

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Henry Ricardo

06/7/2007

Although this 1400-page book stretches the meaning of ‘handbook’ to its breaking point, it is a valuable compendium of information on virtually all aspects of linear algebra and its applications. A similar, but less comprehensive book is Matrix Mathematics by Dennis S. Bernstein, a useful reference work motivated by the utility of “matrix facts” in science, mathematics, and engineering problems.

The book’s 77 chapters are divided into five broad categories: Basic linear algebra, combinatorial matrix theory and graphs, numerical methods, applications, and computational software. There are also six pages of preliminary definitions. Each chapter, written by an expert or team of experts, follows the same basic format: Definitions, Facts, Examples, and References. There are frequent cross references between chapters and sections. The book concludes with a 40-page cross-referenced Glossary, a 9-page Notation Index, and a 56-page Index, beginning with “Abelian, Lie algebras” and ending with “Zyskind-Martin model.” A half-page errata list (as of 3/1/07) and two “additional useful facts” for Section 2.4 are available online.

Unlike the classic two-volume work by Horn and Johnson, this book contains no proofs, but the authors have been encouraged to use standard texts or survey articles as references whenever possible; and most end-of-chapter lists also include reasonably current research articles/reports. (The latest references I caught in my perambulations through this book were a research report published in 2006 and several papers submitted for publication.)

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. Although it is possible that a reader may find some favorite piece of information missing — I couldn’t find the evaluation of a determinant via the *PA* = *LU* decomposition, for example — most users of this volume will be pleased at its thoroughness. Every library that supports mathematics and science departments should have this encyclopedic work on its shelves

Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book *A Modern Introduction to Differential Equations* was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

LINEAR ALGEBRA

BASIC LINEAR ALGEBRA

Vectors, Matrices and Systems of Linear Equations

Linear Independence, Span, and Bases

Linear Transformations

Determinants and Eigenvalues

Inner Product Spaces, Orthogonal Projection, Least Squares and Singular Value Decomposition

MATRICES WITH SPECIAL PROPERTIES

Canonical Forms

Unitary Similarity, Normal Matrices and Spectral Theory

Hermitian and Positive Definite Matrices

Nonnegative and Stochastic Matrices

Partitioned Matrices

ADVANCED LINEAR ALGEBRA

Functions of Matrices

Quadratic, Bilinear and Sesquilinear Forms

Multilinear Algebra

Matrix Equalities and Inequalities

Matrix Perturbation Theory

Pseudospectra

Singular Values and Singular Value Inequalities

Numerical Range

Matrix Stability and Inertia

TOPICS IN ADVANCED LINEAR ALGEBRA

Inverse Eigenvalue Problems

Totally Positive and Totally Nonnegative Matrices

Linear Preserver Problems

Matrices over Integral Domains

Similarity of Families of Matrices

Max-Plus Algebra

Matrices Leaving a Cone Invariant

COMBINATORIAL MATRIX THEORY AND GRAPHS

MATRICES AND GRAPHS

Combinatorial Matrix Theory

Matrices and Graphs

Digraphs and Matrices

Bipartite Graphs and Matrices

TOPICS IN COMBINATORIAL MATRIX THEORY

Permanents

D-Optimal Designs

Sign Pattern Matrices

Multiplicity Lists for the Eigenvalues of Symmetric Matrices with a Given Graph

Matrix Completion Problems

Algebraic Connectivity

NUMERICAL METHODS

NUMERICAL METHODS FOR LINEAR SYSTEMS

Vector and Matrix Norms, Error Analysis, Efficiency and Stability

Matrix Factorizations and Direct Solution of Linear Systems

Least Squares Solution of Linear Systems

Sparse Matrix Methods

Iterative Solution Methods for Linear Systems

NUMERICAL METHODS FOR EIGENVALUES

Symmetric Matrix Eigenvalue Techniques

Unsymmetric Matrix Eigenvalue Techniques

The Implicitly Restarted Arnoldi Method

Computation of the Singular Value Decomposition

Computing Eigenvalues and Singular Values to High Relative Accuracy

COMPUTATIONAL LINEAR ALGEBRA

Fast Matrix Multiplication

Structured Matrix Computations

Large-Scale Matrix Computations

APPLICATIONS

APPLICATIONS TO OPTIMIZATION

Linear Programming

Semidefinite Programming

APPLICATIONS TO PROBABILITY AND STATISTICS

Random Vectors and Linear Statistical Models

Multivariate Statistical Analysis

Markov Chains

APPLICATIONS TO ANALYSIS

Differential Equations and Stability

Dynamical Systems and Linear Algebra

Control Theory

Fourier Analysis

APPLICATIONS TO PHYSICAL AND BIOLOGICAL SCIENCES

Linear Algebra and Mathematical Physics

Linear Algebra in Biomolecular Modeling

APPLICATIONS TO COMPUTER SCIENCE

Coding Theory

Quantum Computation

Information Retrieval and Web Search

Signal Processing

APPLICATIONS TO GEOMETRY

Geometry

Some Applications of Matrices and Graphs in Euclidean Geometry

APPLICATIONS TO ALGEBRA

Matrix Groups

Group Representations

Nonassociative Algebras

Lie Algebras

COMPUTATIONAL SOFTWARE

INTERACTIVE SOFTWARE FOR LINEAR ALGEBRA

MATLAB

Linear Algebra in Maple

Mathematica

PACKAGES OF SUBROUTINES FOR LINEAR ALGEBRA

BLAS

LAPACK

Use of ARPACK and EIGS

Summary of Software for Linear Algebra Freely Available on the Web

GLOSSARY

NOTATION INDEX

INDEX

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