Matrix Algebra as a tool:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Belmont [u.a.]
Duxbury Press
1996
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 212 S. graph. Darst. |
| ISBN: | 0534237126 |
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| 245 | 1 | 0 | |a Matrix Algebra as a tool |c Ali S. Hadi |
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| adam_text | IMAGE 1
MATRIX ALGEBRA
AS A TOOL
ALI S. HADI I CORNELL UNIVERSITY
AN ALEXANDER KUGUSHEV BOOK SUB GOTTLNGEN 7 204 973 309
98A30508
DUXBURY PRESS
AN IMPRINT OF WADSWORTH PUBLISHING COMPANY
)P* AN INTERNATIONAL THOMSON PUBLISHING COMPANY
BELMONT * ALBANY * BONN * BOSTON * CINCINNATI? DETROIT * LONDON * MADRID
* MELBOURNE MEXICO CITY * NEW YORK * PARIS * SAN FRANCISCO * SINGAPORE *
TOKYO * TORONTO * WASHINGTON
IMAGE 2
CONTENTS
PREFACE V
1. INTRODUCTION 1
1.1. WHY MATRIX ALGEBRA? -. . : 1
1.2. SOME DEFINITIONS AND NOTATION 2
1.2.1. WHAT IS A MATRIX? 2
1.2.2. SCALAR ALGEBRA AND MATRIX ALGEBRA 4
1.2.3. MATRIX EQUALITY 5
1.2.4. MATRIX TRANSPOSE 5
1.3. SPECIAL TYPES OF MATRICES 6
1.3.1. SQUARE MATRICES 6
1.3.2. SYMMETRIC MATRICES 6
1.3.3. SKEW-SYMMETRIC MATRICES 7
1.3.4. TRIANGULAR MATRICES . . .. 7
1.3.5. DIAGONAL MATRICES 8
1.3.6. IDENTITY MATRICES 8
1.3.7. UNIT VECTORS 8
1.3.8. MATRICES OF ONES 8
1.3.9. NULL MATRICES 9
1.3.10. CONSTANT MATRICES 9
1.3.11. PARTITIONED MATRICES 9
EXERCISES - 10
2 . SOME MATRIX CALCULATIONS 13
2.1. MATRIX ADDITION AND SUBTRACTION 13
2.2. ELEMENT-WISE PRODUCT 14
2.3. MATRIX INNER PRODUCT 15
2.3.1. DEFINITIONS AND EXAMPLES 15
2.3.2. SPECIAL TYPES OF MATRIX PRODUCTS 18
2.3.3. SOME PROPERTIES OF MATRIX PRODUCTS 21
* 2.4. KRONECKER PRODUCT 23
EXERCISES 26
3 . LINEAR DEPENDENCE AND INDEPENDENCE 31
3.1. INTRODUCTION 31
3.2. LINEAR DEPENDENCE OF TWO VECTORS 32
3.3. LINEAR DEPENDENCE OF A SET OF VECTORS 33
3.4. RANK OF A MATRIX 37
3.5. ELEMENTARY ROW OPERATIONS 38
3.6. ELEMENTARY MATRICES 39
3.7. ELEMENTARY COLUMN OPERATIONS 40
3.8. PERMUTATION MATRICES 40
APPENDIX: REDUCTION TO ROW-ECHELON FORM 41
EXERCISES V 46
IX
IMAGE 3
X CONTENTS
4 . VECTOR GEOMETRY 49
4.1. INTRODUCTION 49
4.2. ALGEBRAIC AND GRAPHIC VIEWS OF VECTORS 49
4.3. GEOMETRIC VIEWS OF VECTORS 49
4.3.1. LENGTH OR MAGNITUDE OF A VECTOR 51
4.3.2. ANGLE BETWEEN TWO VECTORS 52
4.4. VECTOR GEOMETRY AND LINEAR DEPENDENCE 53
4.5. GEOMETRY OF VECTOR ALGEBRA 55
4.5.1. GEOMETRY OF VECTOR ADDITION 55
4.5.2. GEOMETRY OF VECTOR SUBTRACTION 56
4.5.3. GEOMETRY OF MULTIPLYING A VECTOR BY A SCALAR 58
4.5.4. GEOMETRY OF MULTIPLYING A VECTOR BY A MATRIX 61
4.6. VECTOR SPACE 62
EXERCISES 67
5 . THREE MATRIX REDUCTIONS 71
5.1. TRACE, 71
5.2. DETERMINANT 73
5.3. VECTOR NORMS 76
EXERCISES 79
6 . MATRIX INVERSION 81
6.1. ELEMENT-WISE DIVISION 81
6.2. THE REGULAR INVERSE 82
6.2.1. DEFINITIONS 82
6.2.2. EXISTENCE OF MATRIX INVERSE 83
6.2.3. AN ALGORITHM FOR COMPUTING MATRIX INVERSE 84
6.2.4. PROPERTIES OF MATRIX INVERSE 84
* 6.3. GENERALIZED INVERSE 87
6.3.1. DEFINITIONS . . . . 88
6.3.2. EXISTENCE AND UNIQUENESS OF G-INVERSE 88
6.3.3._AN ALGORITHM FOR COMPUTING A G-INVERSE 90
6.3.4. PROPERTIES OF G-INVERSE 91
6.3.5. PARTICULAR TYPES OF G-INVERSE 91
6.3.6. FULL-RANK FACTORIZATION 92
APPENDIX: COMPUTING MATRIX INVERSE 93
EXERCISES 96
7 . LINEAR TRANSFORMATION 101
7.1. INTRODUCTION 101
7.2. DEFINITION 101
* 7.3. ORTHOGONAL ROTATION 106
* 7.4. OBLIQUE ROTATION 108
* 7.5. ORTHOGONAL PROJECTION 108
7.6. SINGULAR AND NONSINGULAR TRANSFORMATIONS 114
EXERCISES * 115
IMAGE 4
CONTENTS XI
8 . SOME APPLICATIONS 117
8.1. SIMULTANEOUS LINEAR EQUATIONS 117
8.1.1. DEFINITIONS 117
8.1.2. GRAPHICAL SOLUTION 118
8.1.3. NUMERICAL SOLUTION 120
8.1.4. THE CASE OF SQUARE COEFFICIENT MATRIX 122
8.1.5. HOMOGENEOUS SYSTEMS 123
8.2. SOME DESCRIPTIVE STATISTICS 123
8.2.1. THE MEAN VECTOR 124
8.2.2. THE VARIANCE-COVARIANCE MATRIX 124
8.2.3. THE CORRELATION MATRIX 129
* 8.3. REGRESSION ANALYSIS 130
8.3.1. INTRODUCTION 130
8.3.2. LEAST SQUARES ESTIMATES 132
8.3.3. AN ILLUSTRATIVE EXAMPLE 133
8.3.4. THE FITTED VALUES AND RESIDUAL VECTORS 134
* 8.4. UPDATING COMPUTATIONS 136
8.4.1. ADDING COLUMNS TO A DATA MATRIX 136
8.4.2. ADDING ROWS TO A DATA MATRIX 138
APPENDIX: SOLVING SYSTEMS OF LINEAR EQUATIONS 140
EXERCISES 142
9 . EIGENVALUES AND EIGENVECTORS 147
9.1. DEFINITION AND DERIVATION 147
9.2. SOME PROPERTIES 151
* 9.3. SINGULAR VALUE DECOMPOSITION 156
EXERCISES 157
1 0 . ELLIPSOIDS AND DISTANCES 161
10.1. ELLIPSOIDS AND SPHERES 161
10.2. DISTANCES 169
- 10.2.1. THE EUCLIDEAN DISTANCE 170
10.2.2. THE ELLIPTICAL DISTANCE 172
* 10.3. SMALLEST ELLIPSOID CONTAINING A SCATTER OF POINTS 177
EXERCISES 181
1 1 . ADDITIONAL APPLICATIONS 185
* 11.1. MATRIXNORMS 185
11.1.1. THE FROBENIUS NORM 185
11.1.2. THE MATRIX P-NORM 186
* 11.2. COLLINEARITY DIAGNOSTICS 187
11.3. CLASSIFICATION OF QUADRATIC FORMS 191
* 11.4. MARKOV CHAINS 194
* 11.5. PRINCIPAL COMPONENT ANALYSIS 197
* 11.6. FACTOR ANALYSIS 199
EXERCISES 201
BIBLIOGRAPHY 205
INDEX * 207
|
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| isbn | 0534237126 |
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| publisher | Duxbury Press |
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| spelling | Hadi, Ali S. Verfasser aut Matrix Algebra as a tool Ali S. Hadi Belmont [u.a.] Duxbury Press 1996 XI, 212 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Matrizenalgebra (DE-588)4139347-8 gnd rswk-swf Matrizenalgebra (DE-588)4139347-8 s DE-188 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022058595&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hadi, Ali S. Matrix Algebra as a tool Matrizenalgebra (DE-588)4139347-8 gnd |
| subject_GND | (DE-588)4139347-8 |
| title | Matrix Algebra as a tool |
| title_auth | Matrix Algebra as a tool |
| title_exact_search | Matrix Algebra as a tool |
| title_full | Matrix Algebra as a tool Ali S. Hadi |
| title_fullStr | Matrix Algebra as a tool Ali S. Hadi |
| title_full_unstemmed | Matrix Algebra as a tool Ali S. Hadi |
| title_short | Matrix Algebra as a tool |
| title_sort | matrix algebra as a tool |
| topic | Matrizenalgebra (DE-588)4139347-8 gnd |
| topic_facet | Matrizenalgebra |
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