Special Matrices and Their Applications in Numerical Mathematics, 2nd Edition (Dover Books on Mathematics Series)

by Miroslav Fiedler

Overview

This revised and corrected second edition of a classic book on special matrices provides researchers in numerical linear algebra and students of general computational mathematics with an essential reference.
Author Miroslav Fiedler, a Professor at the Institute of Computer Science of the Academy of Sciences of the Czech Republic, Prague, begins with definitions of basic concepts of the theory of matrices and fundamental theorems. In subsequent chapters, he explores symmetric and Hermitian matrices, the mutual connections between graphs and matrices, and the theory of entrywise nonnegative matrices. After introducing M-matrices, or matrices of class K, Professor Fiedler discusses important properties of tensor products of matrices and compound matrices and describes the matricial representation of polynomials. He further defines band matrices and norms of vectors and matrices. The final five chapters treat selected numerical methods for solving problems from the field of linear algebra, using the concepts and results explained in the preceding chapters.

Product Details

ISBN-13:	9780486466750
Publisher:	Dover Publications
Publication date:	08/08/2008
Series:	Dover Books on Mathematics Series
Edition description:	2nd ed.
Pages:	384
Product dimensions:	5.40(w) x 8.40(h) x 0.80(d)

Table of Contents

Preface     iii
Basic concepts of matrix theory     1
Matrices     1
Determinants     6
Nonsingular matrices. Inverse matrices     11
Schur complement. Factorization     16
Vector spaces. Rank     22
Eigenvectors, eigenvalues. Characteristic polynomial     25
Similarity. Jordan normal form     27
Symmetric Matrices. Positive Definite and Semidefinite Matrices     41
Euclidean and unitary spaces     41
Symmetric and Hermitian matrices     44
Orthogonal, unitary matrices     45
Gram-Schmidt orthonormalization     50
Positive definite matrices     55
Sylvester's law of inertia     62
Singular value decomposition     64
Graphs and Matrices     71
Digraphs     71
Digraph of a matrix     77
Undirected graphs. Trees     81
Bigraphs     89
Nonnegative Matrices. Stochastic and Doubly Stochastic Matrices     97
Nonnegative matrices     97
The Perron-Frobenius theorem     101
Cyclic matrices     106
Stochastic matrices     117
Doubly stochasticmatrices     120
M-Matrices (Matrices of Classes K and K[subscript 0])     127
Class K     129
Class K[subscript 0]     138
Diagonally dominant matrices     143
Monotone matrices     148
Class P     149
Tensor Product of Matrices. Compound Matrices     157
Tensor product     158
Compound matrices     164
Matrices and polynomials. Stable Matrices     181
Characteristic polynomial     181
Matrices associated with polynomials     184
Bezout matrices     189
Hankel matrices     192
Toeplitz and Lowner matrices     203
Stable matrices     206
Band Matrices     219
Band matrices and graphs     219
Eigenvalues and eigenvectors of tridiagonal matrices     226
Norms and Their Use for Estimation of Eigenvalues     235
Norms     235
Measure of nonsingularity. Dual norms     245
Bounds for eigenvalues     252
Direct Methods for Solving Linear Systems     271
Nonsingular case     271
General case     281
Iterative Methods for Solving Linear Systems      289
General case     289
The Jacobi method     292
The Gauss-Seidel method     294
The SOR method     298
Matrix Inversion     311
Inversion of special matrices     311
The pseudoinverse     318
Numerical Methods for Computing Eigenvalues of Matrices     323
Computation of selected eigenvalues     323
Computation of all the eigenvalues     327
Sparse Matrices     339
Storing. Elimination ordering     339
Envelopes. Profile     348
Bibliography     355
Index     360