Modern aspects of linear algebra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Boca Raton
AMS
1998
|
| Series: | Translations of mathematical monographs
175 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 303 S. graph. Darst. |
| ISBN: | 0821808885 |
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MARC
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| 650 | 7 | |a Forme quadratique |2 jussieu | |
| 650 | 7 | |a Inégalité Weyl |2 jussieu | |
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| adam_text | Contents
Preface xj
Part 1. Introduction 1
Chapter 1. Euclidean Linear Spaces 3
1.1. The simplest properties 3
1.2. Linear mappings and matrices. Determinants 5
1.3. The accuracy problems in computations 9
Chapter 2. Orthogonal and Unitary Linear Transformations 15
2.1. Orthogonal transformations 15
2.2. Orthogonal reflections 16
2.3. Chain of two dimensional rotations 20
Chapter 3. Orthogonal and Unitary Transformations. Singular Values 25
3.1. Representation of a rectangular matrix 25
3.2. Simplification of matrices. Hessenberg matrices 30
3.3. Singular value decomposition 36
3.4. Singular values 40
Part 2. Matrices of Operators in the Euclidean Space 45
Chapter 4. Unitary Similar Transformations. The Schur Theorem 47
4.1. Reduction of a square matrix to the triangular form 47
4.2. The Schur theorem 49
4.3. Criterion for the solvability of a matrix Sylvester equation 50
4.4. Applications of the criterion 53
4.5. Invariant subspaces 56
Chapter 5. Alternation Theorems 59
5.1. Formulation of alternation theorems 59
5.2. The proof of the simplified alternation theorems 62
5.3. The proof of the general alternation theorems 68
5.4. Corollaries of alternation theorems 74
5.5. Useful inequalities for convex functions 78
5.6. Singular values of products of matrices 81
5.7. Foundations of the Sturm method 82
Chapter 6. The Weyl Inequalities 87
6.1. The Weyl inequalities and the Horn theorem 87
vii
viii CONTENTS
6.2. The proof of the Mirsky lemma 93
6.3. Corollaries of the Weyl inequalities 97
Chapter 7. Variational Principles 101
7.1. Stationary values of a Hermitian form on the unit sphere 101
7.2. Stationary values of a Hermitian form 103
7.3. Variational Weber principles 106
7.4. The variational Courant Fischer principle 107
7.5. Inequalities for singular values 108
7.6. Remark about enumeration of singular values 112
7.7. The notion of conditionality for solutions of linear equations 113
7.8. Approximation by matrices of small rank 117
Chapter 8. Resolvent and Dichotomy of Spectrum 123
8.1. Projections onto invariant subspaces 123
8.2. Integral representation of projections 127
8.3. Dichotomy of spectrum 130
8.4. Matrix functions and integral representations 132
8.5. Matrix exponential and matrix powers 135
8.6. Estimate for the resolvent of a matrix 136
Chapter 9. Quadratic Forms in the Spectrum Dichotomy Problem 139
9.1. Integral criteria for the dichotomy quality 139
9.2. Historical remarks 143
9.3. Lyapunov theorems 146
Chapter 10. Matrix Equations and Projections 151
10.1. Solutions to the Lyapunov equations 151
10.2. A generalization of the Lyapunov equation 153
10.3. Matrix pencil regular on the unit circle 159
10.4. Generalization of the discrete Lyapunov equation 163
10.5. Linear and circle dichotomies 170
10.6. Decomposition into invariant subspaces 174
10.7. Remarks about criteria 179
Chapter 11. The Hausdorff Set of a Matrix 183
11.1. The simplest properties of the Hausdorff set 183
11.2. The Hausdorff set of a second order matrix 186
11.3. Geometry of Hausdorff sets and invariant subspaces 193
11.4. Estimates for the resolvent and matrix exponential 201
11.5. Sectorial operators 206
Part 3. Application of Spectral Analysis.
The Most Important Algorithms 213
Chapter 12. Matrix Operators as Models of Differential Operators 215
12.1. A typical example of a sectorial operator 215
12.2. Finite dimensional models of first order operators 221
12.3. Finite dimensional approximations of second order operators 225
12.4. The finite element method 229
CONTKNTS ix
Chapter 13. Application of the Theory of Functions of Complex Variables 235
13.1. The Cart an inequality for polynomials 235
13.2. The Caratheodory inequality 238
13.3. The Jensen inequality 239
13.4. Estimates from below for analytic functions 242
13.5. Criterion for stratification of spectrum 245
13.6. Dependence of the dichotomy criterion on the radius 249
13.7. Logarithmic subharmonicity of the resolvent 253
Chapter 14. Computational Algorithms of Spectral Analysis 257
14.1. The computation of solutions to matrix Lyapunov equations 257
14.2. Computation of the spectrum dichotomy of a regular pencil 260
14.3. The orthogonal elimination algorithm 267
14.4. Properties of the orthogonal elimination algorithm 273
14.5. Approximations of invariant subspaces 283
14.6. Stability of the orthogonal power algorithm 290
14.7. Bases for almost invariant subspaces 294
Bibliography 301
Index 303
|
| adam_txt |
Contents
Preface xj
Part 1. Introduction 1
Chapter 1. Euclidean Linear Spaces 3
1.1. The simplest properties 3
1.2. Linear mappings and matrices. Determinants 5
1.3. The accuracy problems in computations 9
Chapter 2. Orthogonal and Unitary Linear Transformations 15
2.1. Orthogonal transformations 15
2.2. Orthogonal reflections 16
2.3. Chain of two dimensional rotations 20
Chapter 3. Orthogonal and Unitary Transformations. Singular Values 25
3.1. Representation of a rectangular matrix 25
3.2. Simplification of matrices. Hessenberg matrices 30
3.3. Singular value decomposition 36
3.4. Singular values 40
Part 2. Matrices of Operators in the Euclidean Space 45
Chapter 4. Unitary Similar Transformations. The Schur Theorem 47
4.1. Reduction of a square matrix to the triangular form 47
4.2. The Schur theorem 49
4.3. Criterion for the solvability of a matrix Sylvester equation 50
4.4. Applications of the criterion 53
4.5. Invariant subspaces 56
Chapter 5. Alternation Theorems 59
5.1. Formulation of alternation theorems 59
5.2. The proof of the simplified alternation theorems 62
5.3. The proof of the general alternation theorems 68
5.4. Corollaries of alternation theorems 74
5.5. Useful inequalities for convex functions 78
5.6. Singular values of products of matrices 81
5.7. Foundations of the Sturm method 82
Chapter 6. The Weyl Inequalities 87
6.1. The Weyl inequalities and the Horn theorem 87
vii
viii CONTENTS
6.2. The proof of the Mirsky lemma 93
6.3. Corollaries of the Weyl inequalities 97
Chapter 7. Variational Principles 101
7.1. Stationary values of a Hermitian form on the unit sphere 101
7.2. Stationary values of a Hermitian form 103
7.3. Variational Weber principles 106
7.4. The variational Courant Fischer principle 107
7.5. Inequalities for singular values 108
7.6. Remark about enumeration of singular values 112
7.7. The notion of conditionality for solutions of linear equations 113
7.8. Approximation by matrices of small rank 117
Chapter 8. Resolvent and Dichotomy of Spectrum 123
8.1. Projections onto invariant subspaces 123
8.2. Integral representation of projections 127
8.3. Dichotomy of spectrum 130
8.4. Matrix functions and integral representations 132
8.5. Matrix exponential and matrix powers 135
8.6. Estimate for the resolvent of a matrix 136
Chapter 9. Quadratic Forms in the Spectrum Dichotomy Problem 139
9.1. Integral criteria for the dichotomy quality 139
9.2. Historical remarks 143
9.3. Lyapunov theorems 146
Chapter 10. Matrix Equations and Projections 151
10.1. Solutions to the Lyapunov equations 151
10.2. A generalization of the Lyapunov equation 153
10.3. Matrix pencil regular on the unit circle 159
10.4. Generalization of the discrete Lyapunov equation 163
10.5. Linear and circle dichotomies 170
10.6. Decomposition into invariant subspaces 174
10.7. Remarks about criteria 179
Chapter 11. The Hausdorff Set of a Matrix 183
11.1. The simplest properties of the Hausdorff set 183
11.2. The Hausdorff set of a second order matrix 186
11.3. Geometry of Hausdorff sets and invariant subspaces 193
11.4. Estimates for the resolvent and matrix exponential 201
11.5. Sectorial operators 206
Part 3. Application of Spectral Analysis.
The Most Important Algorithms 213
Chapter 12. Matrix Operators as Models of Differential Operators 215
12.1. A typical example of a sectorial operator 215
12.2. Finite dimensional models of first order operators 221
12.3. Finite dimensional approximations of second order operators 225
12.4. The finite element method 229
CONTKNTS ix
Chapter 13. Application of the Theory of Functions of Complex Variables 235
13.1. The Cart an inequality for polynomials 235
13.2. The Caratheodory inequality 238
13.3. The Jensen inequality 239
13.4. Estimates from below for analytic functions 242
13.5. Criterion for stratification of spectrum 245
13.6. Dependence of the dichotomy criterion on the radius 249
13.7. Logarithmic subharmonicity of the resolvent 253
Chapter 14. Computational Algorithms of Spectral Analysis 257
14.1. The computation of solutions to matrix Lyapunov equations 257
14.2. Computation of the spectrum dichotomy of a regular pencil 260
14.3. The orthogonal elimination algorithm 267
14.4. Properties of the orthogonal elimination algorithm 273
14.5. Approximations of invariant subspaces 283
14.6. Stability of the orthogonal power algorithm 290
14.7. Bases for almost invariant subspaces 294
Bibliography 301
Index 303 |
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| index_date | 2024-07-02T13:45:20Z |
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| isbn | 0821808885 |
| language | English Russian |
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| series2 | Translations of mathematical monographs |
| spelling | Godunov, Sergej K. Verfasser aut Modern aspects of linear algebra by S. K. Godunov Boca Raton AMS 1998 XVI, 303 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 175 Algèbre linéaire ram Calcul matriciel jussieu Equation Ljapunov jussieu Forme quadratique jussieu Inégalité Weyl jussieu Méthode élément fini jussieu Opérateurs non auto-adjoints ram Principe variationnel jussieu Résolvante jussieu Théorie spectrale (Mathématiques) ram Théorème Schur jussieu Algebras, Linear Nonselfadjoint operators Spectral theory (Mathematics) Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s DE-604 Translations of mathematical monographs 175 (DE-604)BV000002394 175 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014594820&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Godunov, Sergej K. Modern aspects of linear algebra Translations of mathematical monographs Algèbre linéaire ram Calcul matriciel jussieu Equation Ljapunov jussieu Forme quadratique jussieu Inégalité Weyl jussieu Méthode élément fini jussieu Opérateurs non auto-adjoints ram Principe variationnel jussieu Résolvante jussieu Théorie spectrale (Mathématiques) ram Théorème Schur jussieu Algebras, Linear Nonselfadjoint operators Spectral theory (Mathematics) Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 |
| title | Modern aspects of linear algebra |
| title_auth | Modern aspects of linear algebra |
| title_exact_search | Modern aspects of linear algebra |
| title_exact_search_txtP | Modern aspects of linear algebra |
| title_full | Modern aspects of linear algebra by S. K. Godunov |
| title_fullStr | Modern aspects of linear algebra by S. K. Godunov |
| title_full_unstemmed | Modern aspects of linear algebra by S. K. Godunov |
| title_short | Modern aspects of linear algebra |
| title_sort | modern aspects of linear algebra |
| topic | Algèbre linéaire ram Calcul matriciel jussieu Equation Ljapunov jussieu Forme quadratique jussieu Inégalité Weyl jussieu Méthode élément fini jussieu Opérateurs non auto-adjoints ram Principe variationnel jussieu Résolvante jussieu Théorie spectrale (Mathématiques) ram Théorème Schur jussieu Algebras, Linear Nonselfadjoint operators Spectral theory (Mathematics) Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algèbre linéaire Calcul matriciel Equation Ljapunov Forme quadratique Inégalité Weyl Méthode élément fini Opérateurs non auto-adjoints Principe variationnel Résolvante Théorie spectrale (Mathématiques) Théorème Schur Algebras, Linear Nonselfadjoint operators Spectral theory (Mathematics) Lineare Algebra |
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