Indefinite linear algebra and applications:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2005
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| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XII, 357 S. |
| ISBN: | 3764373490 9783764373498 |
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| 100 | 1 | |a Gohberg, Yiśrāʿēl Z. |d 1928-2009 |e Verfasser |0 (DE-588)118915878 |4 aut | |
| 245 | 1 | 0 | |a Indefinite linear algebra and applications |c Israel Gohberg ; Peter Lancaster ; Leiba Rodman |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2005 | |
| 300 | |a XII, 357 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| _version_ | 1804134590387847168 |
|---|---|
| adam_text | Contents
Preface
vu
1
Introduction and
Outline 1
1.1
Description
of the
Contents
...................... 2
1.2
Notation and Conventions
....................... 3
2
Indefinite Inner
Products
7
2.1
Definition................................
7
2.2
Orthogonality and Orthogonal Bases
................. 9
2.3
Classification of Subspaces
....................... 11
2.4
Exercises
................................ 14
2.5
Notes
.................................. 18
3
Orthogonalization and Orthogonal Polynomials
19
3.1
Regular Orthogonalizations
....................... 19
3.2
The Theorems of
Szegő
and Krein
.................. 27
3.3
One-Step Theorem
........................... 29
3.4
Determinants of
О
ne-
Step Completions
............... 36
3.5
Exercises
................................ 40
3.6
Notes
.................................. 44
4
Classes of Linear Transformations
45
4.1
Adjoint Matrices
............................ 45
4.2
iZ-Selfadjoint Matrices: Examples and Simplest Properties
..... 48
4.3
Н
-Unitary Matrices: Examples and Simplest Properties
...... 50
4.4
A Second Characterization of
Я
-Unitary Matrices
......... 54
4.5
Unitary Similarity
........................... 55
4.6
Contractions
.............................. 57
4.7
Dissipative Matrices
.......................... 59
4.8
Symplectic Matrices
......................... . 62
4.9
Exercises
................................ 66
4.10
Notes
.................................. 72
5
Canonical Forms
73
5.1
Description of a Canonical Form
................... 73
5.2
First Application of the Canonical Form
............... 75
5.3
Proof of Theorem
5.1.1.......................... 77
5.4
Classification of Matrices by Unitary Similarity
........... 82
5.5
Signature Matrices
........................... 85
5.6
Structure of
Я
-Selfadjoint
Matrices
.................. 89
5.7
Я
-Definite Matrices
.......................... 91
5.8
Second Description of the Sign Characteristic
............ 92
5.9
Stability of the Sign Characteristic
.................. 95
5.10
Canonical Forms for Pairs of Hermitian Matrices
.......... 96
5.11
Third Description of the Sign Characteristic
............. 98
5.12
Invariant Maximal
Nonnegative Subspaces
.............. 99
5.13
Inverse Problems
............................ 106
5.14
Canonical Forms for
Я
-Unitaríes:
First Examples
......... 107
5.15
Canonical Forms for
Я
-Unitaries:
General Case
........... 110
5.16
First Applications of the Canonical Form of ii-Unitaries
...... 118
5.17
Further Deductions from the Canonical Form
............ 119
5.18
Exercises
................................ 120
5.19
Notes
.................................. 123
6
Real
Я
-Selfadjoint
Matrices
125
6.1
Real
Я
-Selfadjoint
Matrices and Canonical Forms
......... 125
6.2
Proof of Theorem
6.1.5......................... 128
6.3
Comparison with Results in the Complex Case
........... 131
6.4
Connected Components of Real Unitary Similarity Classes
..... 133
6.5
Connected Components of Real Unitary Similarity Classes
(Я
Fixed)
137
6.6
Exercises
................................ 140
6.7
Notes
.................................. 142
7
Functions of
Я
-Selfadjoint
Matrices
143
7.1
Preliminaries
.............................. 143
7.2
Exponential and Logarithmic Functions
............... 145
7.3
Functions of
Я
-Selfadjoint
Matrices
................. 147
7.4
The Canonical Form and Sign Characteristic
............ 150
7.5
Functions which are Selfadjoint in another Indefinite Inner Product
154
7.6
Exercises
................................ 156
7.7
Notes
.................................. 158
8
Я
-Normal Matrices
159
8.1
Decomposability: First Remarks
................... 159
8.2
Я
-Normal Linear Transformations and Pairs of Commuting Matricesl63
8.3
On Unitary Similarity in an Indefinite Inner Product
........ 165
8.4
The Case of Only One Negative Eigenvalue of
Я
........., 166
8.5
Exercises
................................ 174
8.6
Notes
.................................. 177
9
General Perturbations.
Stability of
Diagonalizable Matrices
179
9.1
General Perturbations
of
Я
-Selfadjoint
Matrices
.......... 179
9.2
Stably
Diagonalizable
Я
-Selfadjoint
Matrices
............ 183
9.3
Analytic
Perturbations
and Eigenvalues
............... 185
9.4
Analytic
Perturbations
and Eigenvectors
............... 189
9.5
The Real Case
............................. 192
9.6
Positive Perturbations
of
Я-
Self
adj
oint Matrices
.......... 193
9.7
Я
-Selfadjoint
Stably r-Diagonalizable
Matrices
........... 195
9.8
General Perturbations
and Stably
Diagonalizable
Я
-Unitary
Matrices
198
9.9
Я
-Unitarily
Stably
it-Diagonalizable Matrices
............ 200
9.10
Exercises
................................ 203
9.11
Notes
.................................. 205
10
Definite
Invariant Subspaces
207
10.1
Semidefinite
and Neutral
Subspaces: A
Particular
H
........ 207
10.2
Plus Matrices and Invariant
Nonnegative
Subspaces
........ 212
10.3
Deductions from Theorem
10.2.4................... 217
10.4
Expansive, Contractive Matrices and Spectral Properties
...... 221
10.5
The Real Case
............................. 226
10.6
Exercises
................................ 227
10.7
Notes
.................................. 228
11
Differential Equations of First Order
229
11.1
Boundedness of solutions
....................... 229
11.2
Hamiltonian Systems of Positive Type with Constant Coefficients
. 232
11.3
Exercises
................................ 234
11.4
Notes
.................................. 236
12
Matrix Polynomials
237
12.1
Standard Pairs and Triples
...................... 238
12.2
Matrix Polynomials with Hermitian Coefficients
........... 242
12.3
Factorization of Hermitian Matrix Polynomials
........... 245
12.4
The Sign Characteristic of Hermitian Matrix Polynomials
..... 249
12.5
The Sign Characteristic of Hermitian Analytic Matrix Functions
. 256
12.6
Hermitian Matrix Polynomials on the Unit Circle
.......... 261
12.7
Exercises
. ............................... 263
12.8
Notes
.................................. 266
13
Differential and Difference Equations of Higher Order
267
13.1
General Solution of a System of Differential Equations
....... 267
13.2
Boundedness for a System of Differential Equations
......... 268
13.3
Stable Boundedness for Differential Equations
............ 270
13.4
The Strongly Hyperbolic Case
..................... 273
13.5
Connected Components of Differential Equations
.......... 274
13.6
A Special Case
............................. 276
13.7
Difference Equations
.......................... 278
13.8
Stable Boundedness for Difference Equations
............ 281
13.9
Connected Components of Difference Equations
........... 284
lS.lOExercises
................................ 286
13.11Notes
.................................. 288
14
Algebraic Riccati Equations
289
14.1
Matrix Pairs in Systems Theory and Control
............ 290
14.2
Origins in Systems Theory
....................... 293
14.3
Preliminaries on the Riccati Equation
................ 295
14.4
Solutions and Invariant Subspaces
.................. 296
14.5
Symmetric Equations
.......................... 297
14.6
An Existence Theorem
......................... 298
14.7
Existence when
M
has Real Eigenvalues
............... 303
14.8
Description of Hermitian Solutions
.................. 307
14.9
Extremal Hermitian Solutions
..................... 309
14.
lOThe CARE with Real Coefficients
................... 312
14.
HThe Concerns of Numerical Analysis
................. 315
14.
^Exercises
................................ 317
.................................. 318
A Topics from Linear Algebra
319
A.I Hermitian Matrices
........................... 319
A.2 The Jordan Form
........................... 321
A.3 Riesz Projections
............................ 332
A.
4
Linear Matrix Equations
........................ 335
A.
5
Perturbation Theory of Subspaces
.................. 335
A.
6
Diagonal Forms for Matrix Polynomials and Matrix Functions
. . . 338
A.
7
Convexity of the Numerical Range
.................. 342
A.8 The Fixed Point Theorem
....................... 344
A.
9
Exercises
................................ 345
Bibliography
349
Index
355
|
| adam_txt |
Contents
Preface
vu
1
Introduction and
Outline 1
1.1
Description
of the
Contents
. 2
1.2
Notation and Conventions
. 3
2
Indefinite Inner
Products
7
2.1
Definition.
7
2.2
Orthogonality and Orthogonal Bases
. 9
2.3
Classification of Subspaces
. 11
2.4
Exercises
. 14
2.5
Notes
. 18
3
Orthogonalization and Orthogonal Polynomials
19
3.1
Regular Orthogonalizations
. 19
3.2
The Theorems of
Szegő
and Krein
. 27
3.3
One-Step Theorem
. 29
3.4
Determinants of
О
ne-
Step Completions
. 36
3.5
Exercises
. 40
3.6
Notes
. 44
4
Classes of Linear Transformations
45
4.1
Adjoint Matrices
. 45
4.2
iZ-Selfadjoint Matrices: Examples and Simplest Properties
. 48
4.3
Н
-Unitary Matrices: Examples and Simplest Properties
. 50
4.4
A Second Characterization of
Я
-Unitary Matrices
. 54
4.5
Unitary Similarity
. 55
4.6
Contractions
. 57
4.7
Dissipative Matrices
. 59
4.8
Symplectic Matrices
. . 62
4.9
Exercises
. 66
4.10
Notes
. 72
5
Canonical Forms
73
5.1
Description of a Canonical Form
. 73
5.2
First Application of the Canonical Form
. 75
5.3
Proof of Theorem
5.1.1. 77
5.4
Classification of Matrices by Unitary Similarity
. 82
5.5
Signature Matrices
. 85
5.6
Structure of
Я
-Selfadjoint
Matrices
. 89
5.7
Я
-Definite Matrices
. 91
5.8
Second Description of the Sign Characteristic
. 92
5.9
Stability of the Sign Characteristic
. 95
5.10
Canonical Forms for Pairs of Hermitian Matrices
. 96
5.11
Third Description of the Sign Characteristic
. 98
5.12
Invariant Maximal
Nonnegative Subspaces
. 99
5.13
Inverse Problems
. 106
5.14
Canonical Forms for
Я
-Unitaríes:
First Examples
. 107
5.15
Canonical Forms for
Я
-Unitaries:
General Case
. 110
5.16
First Applications of the Canonical Form of ii-Unitaries
. 118
5.17
Further Deductions from the Canonical Form
. 119
5.18
Exercises
. 120
5.19
Notes
. 123
6
Real
Я
-Selfadjoint
Matrices
125
6.1
Real
Я
-Selfadjoint
Matrices and Canonical Forms
. 125
6.2
Proof of Theorem
6.1.5. 128
6.3
Comparison with Results in the Complex Case
. 131
6.4
Connected Components of Real Unitary Similarity Classes
. 133
6.5
Connected Components of Real Unitary Similarity Classes
(Я
Fixed)
137
6.6
Exercises
. 140
6.7
Notes'
. 142
7
Functions of
Я
-Selfadjoint
Matrices
143
7.1
Preliminaries
. 143
7.2
Exponential and Logarithmic Functions
. 145
7.3
Functions of
Я
-Selfadjoint
Matrices
. 147
7.4
The Canonical Form and Sign Characteristic
. 150
7.5
Functions which are Selfadjoint in another Indefinite Inner Product
154
7.6
Exercises
. 156
7.7
Notes
. 158
8
Я
-Normal Matrices
159
8.1
Decomposability: First Remarks
. 159
8.2
Я
-Normal Linear Transformations and Pairs of Commuting Matricesl63
8.3
On Unitary Similarity in an Indefinite Inner Product
. 165
8.4
The Case of Only One Negative Eigenvalue of
Я
., 166
8.5
Exercises
. 174
8.6
Notes
. 177
9
General Perturbations.
Stability of
Diagonalizable Matrices
179
9.1
General Perturbations
of
Я
-Selfadjoint
Matrices
. 179
9.2
Stably
Diagonalizable
Я
-Selfadjoint
Matrices
. 183
9.3
Analytic
Perturbations
and Eigenvalues
. 185
9.4
Analytic
Perturbations
and Eigenvectors
. 189
9.5
The Real Case
. 192
9.6
Positive Perturbations
of
Я-
Self
adj
oint Matrices
. 193
9.7
Я
-Selfadjoint
Stably r-Diagonalizable
Matrices
. 195
9.8
General Perturbations
and Stably
Diagonalizable
Я
-Unitary
Matrices
198
9.9
Я
-Unitarily
Stably
it-Diagonalizable Matrices
. 200
9.10
Exercises
. 203
9.11
Notes
. 205
10
Definite
Invariant Subspaces
207
10.1
Semidefinite
and Neutral
Subspaces: A
Particular
H
. 207
10.2
Plus Matrices and Invariant
Nonnegative
Subspaces
. 212
10.3
Deductions from Theorem
10.2.4. 217
10.4
Expansive, Contractive Matrices and Spectral Properties
. 221
10.5
The Real Case
. 226
10.6
Exercises
. 227
10.7
Notes
. 228
11
Differential Equations of First Order
229
11.1
Boundedness of solutions
. 229
11.2
Hamiltonian Systems of Positive Type with Constant Coefficients
. 232
11.3
Exercises
. 234
11.4
Notes
. 236
12
Matrix Polynomials
237
12.1
Standard Pairs and Triples
. 238
12.2
Matrix Polynomials with Hermitian Coefficients
. 242
12.3
Factorization of Hermitian Matrix Polynomials
. 245
12.4
The Sign Characteristic of Hermitian Matrix Polynomials
. 249
12.5
The Sign Characteristic of Hermitian Analytic Matrix Functions
. 256
12.6
Hermitian Matrix Polynomials on the Unit Circle
. 261
12.7
Exercises
. . 263
12.8
Notes
. 266
13
Differential and Difference Equations of Higher Order
267
13.1
General Solution of a System of Differential Equations
. 267
13.2
Boundedness for a System of Differential Equations
. 268
13.3
Stable Boundedness for Differential Equations
. 270
13.4
The Strongly Hyperbolic Case
. 273
13.5
Connected Components of Differential Equations
. 274
13.6
A Special Case
. 276
13.7
Difference Equations
. 278
13.8
Stable Boundedness for Difference Equations
. 281
13.9
Connected Components of Difference Equations
. 284
lS.lOExercises
. 286
13.11Notes
. 288
14
Algebraic Riccati Equations
289
14.1
Matrix Pairs in Systems Theory and Control
. 290
14.2
Origins in Systems Theory
. 293
14.3
Preliminaries on the Riccati Equation
. 295
14.4
Solutions and Invariant Subspaces
. 296
14.5
Symmetric Equations
. 297
14.6
An Existence Theorem
. 298
14.7
Existence when
M
has Real Eigenvalues
. 303
14.8
Description of Hermitian Solutions
. 307
14.9
Extremal Hermitian Solutions
. 309
14.
lOThe CARE with Real Coefficients
. 312
14.
HThe Concerns of Numerical Analysis
. 315
14.
^Exercises
. 317
. 318
A Topics from Linear Algebra
319
A.I Hermitian Matrices
. 319
A.2 The Jordan Form
. 321
A.3 Riesz Projections
. 332
A.
4
Linear Matrix Equations
. 335
A.
5
Perturbation Theory of Subspaces
. 335
A.
6
Diagonal Forms for Matrix Polynomials and Matrix Functions
. . . 338
A.
7
Convexity of the Numerical Range
. 342
A.8 The Fixed Point Theorem
. 344
A.
9
Exercises
. 345
Bibliography
349
Index
355 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gohberg, Yiśrāʿēl Z. 1928-2009 Lancaster, Peter 1929- Rodman, Leiba 1949-2015 |
| author_GND | (DE-588)118915878 (DE-588)123638348 (DE-588)130488631 |
| author_facet | Gohberg, Yiśrāʿēl Z. 1928-2009 Lancaster, Peter 1929- Rodman, Leiba 1949-2015 |
| author_role | aut aut aut |
| author_sort | Gohberg, Yiśrāʿēl Z. 1928-2009 |
| author_variant | y z g yz yzg p l pl l r lr |
| building | Verbundindex |
| bvnumber | BV020866760 |
| classification_rvk | SK 220 |
| ctrlnum | (OCoLC)633796553 (DE-599)BVBBV020866760 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV020866760 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:24:37Z |
| indexdate | 2024-07-09T20:27:02Z |
| institution | BVB |
| isbn | 3764373490 9783764373498 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014188585 |
| oclc_num | 633796553 |
| open_access_boolean | |
| owner | DE-824 DE-20 DE-384 DE-355 DE-BY-UBR DE-83 DE-11 DE-703 DE-188 DE-29T |
| owner_facet | DE-824 DE-20 DE-384 DE-355 DE-BY-UBR DE-83 DE-11 DE-703 DE-188 DE-29T |
| physical | XII, 357 S. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Gohberg, Yiśrāʿēl Z. 1928-2009 Verfasser (DE-588)118915878 aut Indefinite linear algebra and applications Israel Gohberg ; Peter Lancaster ; Leiba Rodman Basel [u.a.] Birkhäuser 2005 XII, 357 S. txt rdacontent n rdamedia nc rdacarrier Skalarproduktraum (DE-588)4181620-1 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Mehrere komplexe Variable (DE-588)4169285-8 gnd rswk-swf Indefinites Skalarprodukt (DE-588)4161471-9 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 s Mehrere komplexe Variable (DE-588)4169285-8 s Skalarproduktraum (DE-588)4181620-1 s Indefinites Skalarprodukt (DE-588)4161471-9 s Lineare Algebra (DE-588)4035811-2 s DE-604 Lancaster, Peter 1929- Verfasser (DE-588)123638348 aut Rodman, Leiba 1949-2015 Verfasser (DE-588)130488631 aut text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2688928&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014188585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gohberg, Yiśrāʿēl Z. 1928-2009 Lancaster, Peter 1929- Rodman, Leiba 1949-2015 Indefinite linear algebra and applications Skalarproduktraum (DE-588)4181620-1 gnd Lineare Algebra (DE-588)4035811-2 gnd Funktion Mathematik (DE-588)4071510-3 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd Indefinites Skalarprodukt (DE-588)4161471-9 gnd |
| subject_GND | (DE-588)4181620-1 (DE-588)4035811-2 (DE-588)4071510-3 (DE-588)4169285-8 (DE-588)4161471-9 |
| title | Indefinite linear algebra and applications |
| title_auth | Indefinite linear algebra and applications |
| title_exact_search | Indefinite linear algebra and applications |
| title_exact_search_txtP | Indefinite linear algebra and applications |
| title_full | Indefinite linear algebra and applications Israel Gohberg ; Peter Lancaster ; Leiba Rodman |
| title_fullStr | Indefinite linear algebra and applications Israel Gohberg ; Peter Lancaster ; Leiba Rodman |
| title_full_unstemmed | Indefinite linear algebra and applications Israel Gohberg ; Peter Lancaster ; Leiba Rodman |
| title_short | Indefinite linear algebra and applications |
| title_sort | indefinite linear algebra and applications |
| topic | Skalarproduktraum (DE-588)4181620-1 gnd Lineare Algebra (DE-588)4035811-2 gnd Funktion Mathematik (DE-588)4071510-3 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd Indefinites Skalarprodukt (DE-588)4161471-9 gnd |
| topic_facet | Skalarproduktraum Lineare Algebra Funktion Mathematik Mehrere komplexe Variable Indefinites Skalarprodukt |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2688928&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014188585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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