Handbook of Linear Algebra

Leslie Hogben

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Description 

  • Presents basic as well as advanced linear algebra concepts, such as matrix perturbation theory and inverse eigenvalue problems
  • Features matrix notation throughout the text
  • Covers combinatorial and numerical linear algebra-two important branches of linear algebra
  • Explores both mathematical and nonmathematical applications, such as quantum computing, control theory, signal processing, and computational biology
  • Discusses software packages useful for linear algebra computations, including MATLAB? Maple? and Mathematica?
  • Provides numerous references for additional information along with a glossary that covers all major linear algebra terminology

    The Handbook of Linear Algebra provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use handbook format. The esteemed international contributors guide you from the very elementary aspects of the subject to the frontiers of current research.

    The book features an accessible layout of parts, chapters, and sections, with each section containing definition, fact, and example segments. The five main parts of the book encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, applications of linear algebra to various mathematical and nonmathematical disciplines, and software packages for linear algebra computations. Within each section, the facts (or theorems) are presented in a list format and include references for each fact to encourage further reading, while the examples illustrate both the definitions and the facts.

    Linearization often enables difficult problems to be estimated by more manageable linear ones, making the Handbook of Linear Algebra essential reading for professionals who deal with an assortment of mathematical problems.

 
Table of Contents
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