Handbook of matrices:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester [u.a.]
Wiley
1996
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 304 S. |
| ISBN: | 0471966886 0471970158 |
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MARC
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| 245 | 1 | 0 | |a Handbook of matrices |c H. Lütkepohl |
| 264 | 1 | |a Chichester [u.a.] |b Wiley |c 1996 | |
| 300 | |a XVI, 304 S. | ||
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Record in the Search Index
| _version_ | 1804125642561683456 |
|---|---|
| adam_text | Contents
Preface
.................................... xi
List of Symbols
.............................. xiii
1
Definitions, Notation, Terminology
................ 1
1.1
Basic Notation and Terminology
................. 1
1.2
Operations Relating Matrices
................... 3
1.3
Inequality Relations Between Matrices
.............. 4
1.4
Operations Related to Individual Matrices
............ 4
1.5
Some Special Matrices
....................... 9
1.6
Some Terms and Quantities Related to Matrices
........ 11
2
Rules for Matrix Operations
.................... 15
2.1
Rules Related to Matrix Sums and Differences
......... 15
2.2
Rules Related to Matrix Multiplication
.............. 16
2.3
Rules Related to Multiplication by a Scalar
........... 18
2.4
Rules for the
Kronecker
Product
................. 19
2.5
Rules for the
Hadamard
Product
................. 20
2.6
Rules for Direct Sums
....................... 22
3
Matrix Valued Functions of a Matrix
.............. 23
3.1
The Transpose
........................... 23
3.2
The Conjugate
........................... 24
3.3
The Conjugate Transpose
..................... 25
3.4
The Adjoint of a Square Matrix
.................. 27
3.5
The Inverse of a Square Matrix
.................. 27
3.5.1
General Results
....................... 27
3.5.2
Inverses Involving Sums and Differences
......... 28
3.5.3
Partitioned Inverses
.................... 29
3.5.4
Inverses Involving Commutation, Duplication and
Elimination Matrices
.................. 31
vi
CONTENTS
3.6
Generalized Inverses
........................ 32
3.6.1
General Results
....................... 32
3.6.2
The Moore-Penrose Inverse
................ 34
3.7
Matrix Powers
........................... 37
3.8
The Absolute Value
........................ 39
4
Trace, Determinant and Rank of a Matrix
........... 41
4.1
The Trace
..............................
41
4.1.1
General Results
....................... 41
4.1.2
Inequalities Involving the Trace
.............. 43
4.1.3
Optimization of Functions Involving the Trace
..... 45
4.2
The Determinant
.......................... 47
4.2.1
General Results
....................... 47
4.2.2
Determinants of Partitioned Matrices
.......... 49
4.2.3
Determinants Involving Duplication Matrices
...... 51
4.2.4
Determinants Involving Elimination Matrices
...... 52
4.2.5
Determinants Involving Both Duplication and Elimina¬
tion Matrices
...................... 53
4.2.6
Inequalities Related to Determinants
........... 54
4.2.7
Optimization of Functions Involving a Determinant
... 56
4.3
The Rank of a Matrix
....................... 58
4.3.1
General Results
....................... 58
4.3.2
Matrix Decompositions Related to the Rank
...... 60
4.3.3
Inequalities Related to the Rank
............. 61
5
Eigenvalues and Singular Values
................. 63
5.1
Definitions
............................. 63
5.2
Properties of Eigenvalues and Eigenvectors
........... 64
5.2.1
General Results
....................... 64
5.2.2
Optimization Properties of Eigenvalues
......... 67
5.2.3
Matrix Decompositions Involving Eigenvalues
...... 69
5.3
Eigenvalue Inequalities
....................... 72
5.3.1
Inequalities for the Eigenvalues of a Single Matrix
... 72
5.3.2
Relations Between Eigenvalues of More Than One Matrix
74
5.4
Results for the Spectral Radius
.................. 76
5.5
Singular Values
........................... 78
5.5.1
General Results
....................... 78
5.5.2
Inequalities
......................... 80
6
Matrix Decompositions and Canonical Forms
......... 83
6.1
Complex Matrix Decompositions
................. 83
6.1.1
Jordan Type Decompositions
............... 83
6.1.2
Diagonal Decompositions
................. 85
CONTENTS vü
6.1.3
Other
Triangular
Decompositions and Factorizations
. . 86
6.1.4
Miscellaneous Decompositions
............... 88
6.2
Real Matrix Decompositions
.................... 89
6.2.1
Jordan Decompositions
.................. 89
6.2.2
Other Real Block Diagonal and Diagonal Decompositions
90
6.2.3
Other Triangular and Miscellaneous Reductions
..... 92
7
Vectorization Operators
....................... 95
7.1
Definitions
.............................. 95
7.2
Rules for the
vec
Operator
..................... 97
7.3
Rules for the vech Operator
.................... 99
8
Vector and Matrix Norms
.....................101
8.1
General Definitions
.........................101
8.2
Specific Norms and Inner Products
................103
8.3
Results for General Norms and Inner Products
.........104
8.4
Results for Matrix Norms
.....................106
8.4.1
General Matrix Norms
...................106
8.4.2
Induced Matrix Norms
...................108
8.5
Properties of Special Norms
....................109
8.5.1
General Results
.......................109
8.5.2
Inequalities
.........................
Ill
9
Properties of Special Matrices
...................113
9.1
Circulant
Matrices
.........................113
9.2
Commutation Matrices
.......................115
9.2.1
General Properties
.....................116
9.2.2 Kronecker
Products
....................117
9.2.3
Relations With Duplication and Elimination Matrices
. 118
9.3
Convergent Matrices
........................119
9.4
Diagonal Matrices
.........................120
9.5
Duplication Matrices
........................122
9.5.1
General Properties
.....................122
9.5.2
Relations With Commutation and Elimination Matrices
123
9.5.3
Expressions With
vec
and vech Operators
........123
9.5.4
Duplication Matrices and
Kronecker
Products
......124
9.5.5
Duplication Matrices, Elimination Matrices and Kro-
necker Products
.................... 126
9.6
Elimination Matrices
........................127
9.6.1
General Properties
.....................127
9.6.2
Relations With Commutation and Duplication Matrices
127
9.6.3
Expressions With
vec
and vech Operators
........128
9.6.4
Elimination Matrices and
Kronecker
Products
......128
viii CONTENTS
9.6.5
Elimination Matrices, Duplication Matrices and Kro-
necker Products
.................... 130
9.7
Hermitian Matrices
.........................131
9.7.1
General Results
....................... 131
9.7.2
Eigenvalues of Hermitian Matrices
............ 133
9.7.3
Eigenvalue Inequalities
................... 134
9.7.4
Decompositions of Hermitian Matrices
.......... 137
9.8
Idempotent Matrices
........................ 138
9.9
Nonnegative, Positive
and Stochastic Matrices
.......... 139
9.9.1
Definitions
......................... 139
9.9.2
General Results
....................... 140
9.9.3
Results Related to the Spectral Radius
.......... 141
9.10
Orthogonal Matrices
........................ 142
9.10.1
General Results
.......................143
9.10.2
Decompositions of Orthogonal Matrices
.........144
9.11
Partitioned Matrices
........................144
9.11.1
General Results
....................... 145
9.11.2
Determinants of Partitioned Matrices
.......... 146
9.11.3
Partitioned Inverses
.................... 147
9.11.4
Partitioned Generalized Inverses
............. 148
9.11.5
Partitioned Matrices Related to Duplication Matrices
. 149
9.12
Positive Definite, Negative Definite and
Semidefinite
Matrices
. 150
9.12.1
General Properties
.....................151
9.12.2
Eigenvalue Results
.....................153
9.12.3
Decomposition Theorems for Definite Matrices
.....154
9.13
Symmetric Matrices
........................156
9.13.1
General Properties
..................... 156
9.13.2
Symmetry and Duplication Matrices
........... 157
9.13.3
Eigenvalues of Symmetric Matrices
............ 158
9.13.4
Eigenvalue Inequalities
................... 159
9.13.5
Decompositions of Symmetric and Skew-Symmetric
Matrices
......................... 163
9.14
Triangular Matrices
........................164
9.14.1
Properties of General Triangular Matrices
........ 164
9.14.2
Triangularity, Elimination and Duplication Matrices
. . 165
9.14.3
Properties of Strictly Triangular Matrices
........ 167
9.15
Unitary Matrices
.......................... 167
10
Vector and Matrix Derivatives
..................171
10.1
Notation
...............................171
10.2
Gradients and Hessian Matrices of Real Valued Functions with
Vector Arguments
....................... 174
10.2.1
Gradients
..........................174
CONTENTS ix
10.2.2
Hessian Matrices
......................175
10.3
Derivatives of Real Valued Functions with Matrix Arguments
. 176
10.3.1
General and Miscellaneous Rules
............. 176
10.3.2
Derivatives of the Trace
.................. 177
10.3.3
Derivatives of Determinants
................ 181
10.4
J
acobian Matrices of Linear Functions
.............. 183
10.4.1
Linear Functions with General Matrix Arguments
. . . 183
10.4.2
Linear Functions with Symmetric Matrix Arguments
. . 185
10.4.3
Linear Functions with Triangular Matrix Arguments
. . 187
10.4.4
Linear Functions of Vector and Matrix Valued Func¬
tions with Vector Arguments
............. 188
10.5
Product Rules
............................189
10.5.1
Matrix Products
...................... 189
10.5.2 Kronecker
and
Hadamard
Products
........... 192
10.5.3
Functions with Symmetric Matrix Arguments
...... 193
10.5.4
Functions with Lower Triangular Matrix Arguments
. . 195
10.5.5
Products of Matrix Valued Functions with Vector
Arguments
....................... 196
10.6
Jacobian Matrices of Functions Involving Inverse Matrices
. . . 198
10.6.1
Matrix Products
...................... 198
10.6.2 Kronecker
and
Hadamard
Products
........... 199
10.6.3
Matrix Valued Functions with Vector Arguments
.... 200
10.7
Chain Rules and Miscellaneous Jacobian Matrices
....... 202
10.8
Jacobian Determinants
....................... 204
10.8.1
Linear Transformations
..................204
10.8.2
Nonlinear Transformations
................ 206
10.9
Matrix Valued Functions of a Scalar Variable
..........208
11
Polynomials, Power Series and Matrices
............211
11.1
Definitions and Notations
.....................211
11.1.1
Definitions and Notation Related to Polynomials
. . . .211
11.1.2
Matrices Related to Polynomials
.............213
11.1.3
Polynomials and Power Series Related to Matrices
. . . 214
11.2
Results Relating Polynomials and Matrices
...........215
11.3
Polynomial Matrices
........................217
11.3.1
Definitions
.........................217
11.3.2
Results for Polynomial Matrices
.............221
Appendix A Dictionary of Matrices and Related Terms
.... 225
References
..................................287
Index
.....................................291
|
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| classification_rvk | QH 110 QH 140 SK 220 |
| classification_tum | MAT 150b |
| ctrlnum | (OCoLC)35447455 (DE-599)BVBBV011148836 |
| dewey-full | 512.9/434 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 |
| dewey-search | 512.9/434 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV011148836 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:04:48Z |
| institution | BVB |
| isbn | 0471966886 0471970158 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007472960 |
| oclc_num | 35447455 |
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| physical | XVI, 304 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Wiley |
| record_format | marc |
| spelling | Lütkepohl, Helmut 1951- Verfasser (DE-588)10979544X aut Handbook of matrices H. Lütkepohl Chichester [u.a.] Wiley 1996 XVI, 304 S. txt rdacontent n rdamedia nc rdacarrier Matrices gtt Matrices Matrizenrechnung (DE-588)4126963-9 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s DE-604 Matrizenrechnung (DE-588)4126963-9 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007472960&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lütkepohl, Helmut 1951- Handbook of matrices Matrices gtt Matrices Matrizenrechnung (DE-588)4126963-9 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4126963-9 (DE-588)4037968-1 |
| title | Handbook of matrices |
| title_auth | Handbook of matrices |
| title_exact_search | Handbook of matrices |
| title_full | Handbook of matrices H. Lütkepohl |
| title_fullStr | Handbook of matrices H. Lütkepohl |
| title_full_unstemmed | Handbook of matrices H. Lütkepohl |
| title_short | Handbook of matrices |
| title_sort | handbook of matrices |
| topic | Matrices gtt Matrices Matrizenrechnung (DE-588)4126963-9 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrices Matrizenrechnung Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007472960&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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