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Miroslav Fiedler, "Special Matrices and Their Applications in Numerical Mathematics"

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 Löwner 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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