Handbook of Linear Algebra

Leslie Hogben, editor

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

BLL Rating:

BLL**

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

MAA Review
Table of Contents

[Reviewed by

Henry Ricardo

, on

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 ([email protected]) 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