A combinatorial approach to matrix theory and its applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2009
|
| Series: | Discrete mathematics and its applications
52 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 267 S. Ill. |
| ISBN: | 9781420082234 |
Staff View
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| 100 | 1 | |a Brualdi, Richard A. |d 1939- |e Verfasser |0 (DE-588)113645112 |4 aut | |
| 245 | 1 | 0 | |a A combinatorial approach to matrix theory and its applications |c Richard A. Brualdi ; Dragoš Cvetković |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2009 | |
| 300 | |a XIII, 267 S. |b Ill. | ||
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| 490 | 1 | |a Discrete mathematics and its applications |v 52 | |
| 500 | |a Includes bibliographical references and index | ||
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Record in the Search Index
| _version_ | 1804137979338293248 |
|---|---|
| adam_text | Contents
Preface
xi
Dedication
xv
1
Introduction
1
1.1
Graphs
......................... 2
1.2
Digraphs
......................... 8
1.3
Some Classical Combinatorics
............ 10
1.4
Fields
......................... 13
1.5
Vector Spaces
..................... 17
1.6
Exercises
........................ 23
2
Basic Matrix Operations
27
2.1
Basic Concepts
.................... 27
2.2
The
König
Digraph of a Matrix
........... 35
2.3
Partitioned Matrices
................. 43
2.4
Exercises
........................ 46
3
Powers of Matrices
49
3.1
Matrix Powers and Digraphs
............. 49
3.2
Circulant
Matrices
.................. 58
3.3
Permutations with Restrictions
........... 59
3.4
Exercises
........................ 60
4
Determinants
63
4.1
Definition of the Determinant
............ 63
4.2
Properties of Determinants
.............. 72
4.3
A Special Determinant Formula
........... 85
vn
viii CONTENTS
4.4
Classical Definition of the Determinant
....... 87
4.5
Laplace Determinant Development
......... 91
4.6
Exercises
........................ 94
5
Matrix Inverses
97
5.1
Adjoint and Its Determinant
............. 97
5.2
Inverse of a Square Matrix
.............. 101
5.3
Graph-Theoretic Interpretation
........... 103
5.4
Exercises
........................ 107
6
Systems of Linear Equations
109
6.1
Solutions of Linear Systems
............. 109
6.2
Cramer s Formula
................... 118
6.3
Solving Linear Systems by Digraphs
......... 121
6.4
Signal Flow Digraphs of Linear Systems
...... 127
6.5
Sparse Matrices
.................... 133
6.6
Exercises
........................ 137
7
Spectrum of a Matrix
139
7.1
Eigenvectors and Eigenvalues
............ 139
7.2
The Cayley-Hamilton Theorem
........... 147
7.3
Similar Matrices and the JCF
............ 150
7.4
Spectrum of
Circulants
................ 165
7.5
Exercises
........................ 167
8
Nonnegative
Matrices
171
8.1
Irreducible and Reducible Matrices
......... 171
8.2
Primitive and
Imprimitive
Matrices
......... 174
8.3
The Perron-Frobenius Theorem
........... 179
8.4
Graph Spectra
..................... 184
8.5
Exercises
........................ 188
9
Additional Topics
191
9.1
Tensor and
Hadamard
Product
........... 191
9.2
Eigenvalue Inclusion Regions
............. 196
9.3
Permanent and SNS-Matrices
............ 204
9.4
Exercises
........................ 215
CONTENTS ix
10 Applications 217
10.1
Electrical
Engineering:
Flow Graphs
........218
10.2
Physics: Vibration of a Membrane
..........224
10.3
Chemistry: Unsaturated Hydrocarbons
.......229
10.4
Exercises
........................240
Coda
241
Answers and Hints
245
Bibliography
253
Index
261
|
| adam_txt |
Contents
Preface
xi
Dedication
xv
1
Introduction
1
1.1
Graphs
. 2
1.2
Digraphs
. 8
1.3
Some Classical Combinatorics
. 10
1.4
Fields
. 13
1.5
Vector Spaces
. 17
1.6
Exercises
. 23
2
Basic Matrix Operations
27
2.1
Basic Concepts
. 27
2.2
The
König
Digraph of a Matrix
. 35
2.3
Partitioned Matrices
. 43
2.4
Exercises
. 46
3
Powers of Matrices
49
3.1
Matrix Powers and Digraphs
. 49
3.2
Circulant
Matrices
. 58
3.3
Permutations with Restrictions
. 59
3.4
Exercises
. 60
4
Determinants
63
4.1
Definition of the Determinant
. 63
4.2
Properties of Determinants
. 72
4.3
A Special Determinant Formula
. 85
vn
viii CONTENTS
4.4
Classical Definition of the Determinant
. 87
4.5
Laplace Determinant Development
. 91
4.6
Exercises
. 94
5
Matrix Inverses
97
5.1
Adjoint and Its Determinant
. 97
5.2
Inverse of a Square Matrix
. 101
5.3
Graph-Theoretic Interpretation
. 103
5.4
Exercises
. 107
6
Systems of Linear Equations
109
6.1
Solutions of Linear Systems
. 109
6.2
Cramer's Formula
. 118
6.3
Solving Linear Systems by Digraphs
. 121
6.4
Signal Flow Digraphs of Linear Systems
. 127
6.5
Sparse Matrices
. 133
6.6
Exercises
. 137
7
Spectrum of a Matrix
139
7.1
Eigenvectors and Eigenvalues
. 139
7.2
The Cayley-Hamilton Theorem
. 147
7.3
Similar Matrices and the JCF
. 150
7.4
Spectrum of
Circulants
. 165
7.5
Exercises
. 167
8
Nonnegative
Matrices
171
8.1
Irreducible and Reducible Matrices
. 171
8.2
Primitive and
Imprimitive
Matrices
. 174
8.3
The Perron-Frobenius Theorem
. 179
8.4
Graph Spectra
. 184
8.5
Exercises
. 188
9
Additional Topics
191
9.1
Tensor and
Hadamard
Product
. 191
9.2
Eigenvalue Inclusion Regions
. 196
9.3
Permanent and SNS-Matrices
. 204
9.4
Exercises
. 215
CONTENTS ix
10 Applications 217
10.1
Electrical
Engineering:
Flow Graphs
.218
10.2
Physics: Vibration of a Membrane
.224
10.3
Chemistry: Unsaturated Hydrocarbons
.229
10.4
Exercises
.240
Coda
241
Answers and Hints
245
Bibliography
253
Index
261 |
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| isbn | 9781420082234 |
| language | English |
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| spelling | Brualdi, Richard A. 1939- Verfasser (DE-588)113645112 aut A combinatorial approach to matrix theory and its applications Richard A. Brualdi ; Dragoš Cvetković Boca Raton [u.a.] CRC Press 2009 XIII, 267 S. Ill. txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications 52 Includes bibliographical references and index Matrices Combinatorial analysis Kombinatorische Analysis (DE-588)4164746-4 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s Kombinatorische Analysis (DE-588)4164746-4 s DE-604 Cvetković, Dragoš M. 1941- Sonstige (DE-588)110230485 oth Discrete mathematics and its applications 52 (DE-604)BV023551867 52 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016712215&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brualdi, Richard A. 1939- A combinatorial approach to matrix theory and its applications Discrete mathematics and its applications Matrices Combinatorial analysis Kombinatorische Analysis (DE-588)4164746-4 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4164746-4 (DE-588)4037968-1 |
| title | A combinatorial approach to matrix theory and its applications |
| title_auth | A combinatorial approach to matrix theory and its applications |
| title_exact_search | A combinatorial approach to matrix theory and its applications |
| title_exact_search_txtP | A combinatorial approach to matrix theory and its applications |
| title_full | A combinatorial approach to matrix theory and its applications Richard A. Brualdi ; Dragoš Cvetković |
| title_fullStr | A combinatorial approach to matrix theory and its applications Richard A. Brualdi ; Dragoš Cvetković |
| title_full_unstemmed | A combinatorial approach to matrix theory and its applications Richard A. Brualdi ; Dragoš Cvetković |
| title_short | A combinatorial approach to matrix theory and its applications |
| title_sort | a combinatorial approach to matrix theory and its applications |
| topic | Matrices Combinatorial analysis Kombinatorische Analysis (DE-588)4164746-4 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrices Combinatorial analysis Kombinatorische Analysis Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016712215&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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