Special Matrices and Their Applications in Numerical Mathematics Second Edition

by Fiedler, Miroslav

• ISBN13: 9780486466750
• ISBN10: 0486466752
• Edition: 2nd
• Format: Paperback
• Copyright: 2008-08-08
• Publisher: Dover Publications

Summary

This revised edition of a classic on special matrices provides researchers in numerical linear algebra and students of general computational mathematics with an essential reference. Its final section uses concepts and results of numerical methods explained in the preceding chapters to solve problems of linear algebra. 1986 edition.

Table of Contents

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