Special Matrices and Their Applications in Numerical Mathematics - ISBN:9024729572

Special Matrices and Their Applications in Numerical Mathematics

Date: 15 April 2011, 12:14

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Contents
1. Basic Concepts of the Theory of Matrices 5
Matrices 5
Determinants 10
Nonsingular matrices. Inverse matrices 15
Schur complement. Factorization 19
Vector spaces. Rank 24
Eigenvectors, eigenvalues. Characteristic polynomial 26
Similarity. Jordan normal form 28
Exercises 36
2. Symmetric Matrices. Positive Definite and Semidefinite Matrices 39
Euclidean and unitary spaces 39
Symmetric and Hermitian matrices 41
Orthogonal and unitary matrices 43
Gram-Schmidt orthonormalization. Schur's theorem 47
Positive definite and positive semidefinite matrices 51
Sylvester's law of inertia 57
Singular value decomposition 59
Exercises 63
3. Graphs and Matrices 65
Digraphs 65
Digraph of a matrix 70
Undirected graphs. Trees 73
Bigraphs 80
Exercises 85
4. Nonnegative Matrices. Stochastic and Doubly Stochastic Matrices 87
Nonnegative matrices 87
The Perron-Frobenius theorem 90
Cyclic matrices 95
Stochastic matrices 105
Doubly stochastic matrices 107
Exercises 111
5. M-Matrices (Matrices of Classes K and Ko) 112
Class K 114
Class Ko 121
Diagonally dominant matrices 126
Monotone matrices 130
Class P 131
Exercises 135
6. Tensor Product of Matrices. Compound Matrices 136
Tensor product 137
Compound matrices 142
Exercises 155
7. Matrices and Polynomials. Stable matrices 157
Characteristic polynomial 157
Matrices associated with polynomials 160
Bezout matrices 164
Hankel matrices 167
Toeplitz and Lowner matrices 177
Stable matrices 178
Exercises 186
8. Band Matrices 189
Band matrices and graphs 189
Eigenvalues and eigenvectors of tridiagonal matrices 195
Exercises 200
9. Norms and Their Use for Estimation of Eigenvalues 201
Norms 201
Measure of nonsingularity. Dual norms 209
Bounds for eigenvalues 215
Exercises 230
10. Direct Methods for Solving Linear Systems 231
Nonsingular case 231
General case 239
Exercises 244
11. Iterative Methods for Solving Linear Systems 245
The Jacobi method 247
The Gauss-Seidel method 249
The SOR method 252
Exercises 262
12. Matrix Inversion 264
Inversion of special matrices 264
The pseudo inverse 270
Exercises 271
13. Numerical Methods for Computing Eigenvalues of Matrices 273
Computation of selected eigenvalues 273
Computation of all the eigenvalues 276
Exercises 284
14. Sparse matrices 286
Storing. Elimination ordering 286
Envelopes. Profile 294
Exercises 298
Bibliography 299
Subject Index 303

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