# Handbook of Linear Algebra

## Preview

## 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 GraphsCombinatorial 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 MethodsNumerical 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 DisciplinesApplications 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 SoftwareInteractive 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

## 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."

—EMS Newsletter