Matrix algebra and its applications to statistics and econometrics:
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| Format: | Buch |
| Sprache: | English |
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New Jersey[u.a.]
World Scientific
2007
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| Ausgabe: | Reprint. |
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| Beschreibung: | XIX, 535 S. graph. Darst. |
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| adam_text | Titel: Matrix algebra and its applications to statistics and econometrics
Autor: Rao, Calyampudi Radhakrishna
Jahr: 2007
CONTENTS
Preface ......................................................... vii
Notation........................................................ xi
CHAPTER 1. VECTOR SPACES
1.1 Rings and Fields.............................................. 1
1.2 Mappings................................................... 14
1.3 Vector Spaces............................................... 16
1.4 Linear Independence and Basis of a Vector Space................ 19
1.5 Subspaces .................................................. 24
1.6 Linear Equations............................................ 29
1.7 Dual Space ................................................. 35
1.8 Quotient Space.............................................. 41
1.9 Projective Geometry......................................... 42
CHAPTER 2. UNITARY AND EUCLIDEAN SPACES
2.1 Inner Product............................................... 51
2.2 Orthogonality............................................... 56
2.3 Linear Equations........................¦.................... 66
2.4 Linear Functional........................................... 71
2.5 Semi-inner Product.......................................... 76
2.6 Spectral Theory............................................. 83
2.7 Conjugate Bilinear Functionals and Singular Value
Decomposition............................................. 101
CHAPTER 3. LINEAR TRANSFORMATIONS AND
MATRICES
3.1 Preliminaries............................................... 107
3.2 Algebra of Transformations.................................. 110
3.3 Inverse Transformations..................................... 116
3.4 Matrices................................................... 120
xvi MATRIX ALGERBA THEORY AND APPLICATIONS
CHAPTER 4. CHARACTERISTICS OF MATRICES
4.1 Rank and Nullity of a Matrix ................................ 128
4.2 Rank and Product of Matrices................................ 131
4.3 Rank Factorization and Further Results....................... 136
4.4 Determinants .............................................. 142
4.5 Determinants and Minors.................................... 146
CHAPTER 5. FACTORIZATION OF MATRICES
5.1 Elementary Matrices........................................ 157
5.2 Reduction of General Matrices............................... 160
5.3 Factorization of Matrices with Complex Entries................. 166
5.4 Eigenvalues and Eigenvectors................................. 177
5.5 Simultaneous Reduction of Two Matrices...................... 184
5.6 A Review of Matrix Factorizations............................ 188
CHAPTER 6. OPERATIONS ON MATRICES
6.1 Kronecker Product.......................................... 193
6.2 The Vec Operation.......................................... 200
6.3 The Hadamard-Schur Product................................ 203
6.4 Khatri-Rao Product......................................... 216
6.5 Matrix Derivatives.......................................... 223
CHAPTER 7. PROJECTORS AND IDEMPOTENT
OPERATORS
7.1 Projectors................................................. 239
7.2 Invariance and Reducibility.................................. 245
7.3 Orthogonal Projection....................................... 248
7.4 Idempotent Matrices........................................ 250
7.5 Matrix Representation of Projectors........................... 256
CHAPTER 8. GENERALIZED INVERSES
8.1 Right and Left Inverses...................................... 264
8.2 Generalized Inverse (y-inverse) ............................... 265
8.3 Geometric Approach: LMN-inverse ........................... 282
8.4 Minimum Norm Solution.................................... 288
Contents xvii
8.5 Least Squares Solution...................................... 289
8.6 Minimum Norm Least Squares Solution........................ 291
8.7 Various Types of «/-inverses.................................. 292
8.8 G-inverses Through Matrix Approximations.................... 296
8.9 Gauss-Markov Theorem..................................... 300
CHAPTER 9. MAJORIZATION
9.1 Majorization............................................... 303
9.2 A Gallery of Functions...................................... 307
9.3 Basic Results............................................... 308
CHAPTER 10. INEQUALITIES FOR EIGENVALUES
10.1 Monotonicity Theorem.................................... 322
10.2 Interlace Theorems ....................................... 328
10.3 Courant-Fischer Theorem.................................. 332
10.4 Poincaré Separation Theorem.............................. 337
10.5 Singular Values and Eigenvalues............................ 339
10.6 Products of Matrices, Singular Values, and Horn s
Theorem ................................................ 340
10.7 Von Neumann s Theorem.................................. 342
CHAPTER 11. MATRIX APPROXIMATIONS
11.1 Norm on a Vector Space................................... 361
11.2 Norm on Spaces of Matrices............. ................... 363
11.3 Unitarily Invariant Norms ................................. 374
11.4 Some Matrix Optimization Problems........................ 383
11.5 Matrix Approximations.................................... 388
11.6 M, N-invariant Norm and Matrix Approximations............. 394
11.7 Fitting a Hyperplane to a Set of Points...................... 398
CHAPTER 12. OPTIMIZATION PROBLEMS IN
STATISTICS AND ECONOMETRICS
12.1 Linear Models............................................ 403
12.2 Some Useful Lemmas...................................... 403
12.3 Estimation in a Linear Model.............................. 406
12.4 A Trace Minimization Problem............................. 409
12.5 Estimation of Variance.................................... 413
xviii MATRIX ALGEBRA THEORY AND APPLICATIONS
12.6 The Method of MINQUE: A Prologue....................... 415
12.7 Variance Components Models and Unbiased Estimation ....... 416
12.8 Normality Assumption and Invariant Estimators.............. 419
12.9 The Method of MINQUE.................................. 422
12.10 Optimal Unbiased Estimation.............................. 425
12.11 Total Least Squares....................................... 428
CHAPTER 13. QUADRATIC SUBSPACES
13.1 Basic Ideas .............................................. 433
13.2 The Structure of Quadratic Subspaces....................... 438
13.3 Commutators of Quadratic Subspaces....................... 442
13.4 Estimation of Variance Components......................... 443
CHAPTER 14. INEQUALITIES WITH APPLICATIONS
IN STATISTICS
14.1 Some Results on nnd and pd Matrices....................... 449
14.2 Cauchy-Schwartz and Related Inequalities................... 454
14.3 Hadamard Inequality...................................... 456
14.4 Holder s Inequality........................................ 457
14.5 Inequalities in Information Theory.......................... 458
14.6 Convex Functions and Jensen s Inequality.................... 459
14.7 Inequalities Involving Moments............................. 461
14.8 Kantorovich Inequality and Extensions...................... 462
CHAPTER 15. NON-NEGATIVE MATRICES
15.1 Perron-ïïobenius Theorem................................. 467
15.2 Leontief Models in Economics.............................. 477
15.3 Markov Chains........................................... 481
15.4 Genetic Models........................................... 485
15.5 Population Growth Models................................. 489
CHAPTER 16. MISCELLANEOUS COMPLEMENTS
16.1 Simultaneous Decomposition of Matrices..................... 493
16.2 More on Inequalities...................................... 494
16.3 Miscellaneous Results on Matrices.......................... 497
16.4 Toeplitz Matrices......................................... 501
16.5 Restricted Eigenvalue Problem............................. 506
Contents xix
16.6 Product of Two Raleigh Quotients.......................... 507
16.7 Matrix Orderings and Projection ........................... 508
16.8 Soft Majorization......................................... 509
16.9 Circulants............................................... 511
16.10 Hadamard Matrices....................................... 514
16.11 Miscellaneous Exercises.................................... 515
REFERENCES................................................. 519
INDEX........................................................ 529
|
| adam_txt |
Titel: Matrix algebra and its applications to statistics and econometrics
Autor: Rao, Calyampudi Radhakrishna
Jahr: 2007
CONTENTS
Preface . vii
Notation. xi
CHAPTER 1. VECTOR SPACES
1.1 Rings and Fields. 1
1.2 Mappings. 14
1.3 Vector Spaces. 16
1.4 Linear Independence and Basis of a Vector Space. 19
1.5 Subspaces . 24
1.6 Linear Equations. 29
1.7 Dual Space . 35
1.8 Quotient Space. 41
1.9 Projective Geometry. 42
CHAPTER 2. UNITARY AND EUCLIDEAN SPACES
2.1 Inner Product. 51
2.2 Orthogonality. 56
2.3 Linear Equations.¦. 66
2.4 Linear Functional. 71
2.5 Semi-inner Product. 76
2.6 Spectral Theory. 83
2.7 Conjugate Bilinear Functionals and Singular Value
Decomposition. 101
CHAPTER 3. LINEAR TRANSFORMATIONS AND
MATRICES
3.1 Preliminaries. 107
3.2 Algebra of Transformations. 110
3.3 Inverse Transformations. 116
3.4 Matrices. 120
xvi MATRIX ALGERBA THEORY AND APPLICATIONS
CHAPTER 4. CHARACTERISTICS OF MATRICES
4.1 Rank and Nullity of a Matrix . 128
4.2 Rank and Product of Matrices. 131
4.3 Rank Factorization and Further Results. 136
4.4 Determinants . 142
4.5 Determinants and Minors. 146
CHAPTER 5. FACTORIZATION OF MATRICES
5.1 Elementary Matrices. 157
5.2 Reduction of General Matrices. 160
5.3 Factorization of Matrices with Complex Entries. 166
5.4 Eigenvalues and Eigenvectors. 177
5.5 Simultaneous Reduction of Two Matrices. 184
5.6 A Review of Matrix Factorizations. 188
CHAPTER 6. OPERATIONS ON MATRICES
6.1 Kronecker Product. 193
6.2 The Vec Operation. 200
6.3 The Hadamard-Schur Product. 203
6.4 Khatri-Rao Product. 216
6.5 Matrix Derivatives. 223
CHAPTER 7. PROJECTORS AND IDEMPOTENT
OPERATORS
7.1 Projectors. 239
7.2 Invariance and Reducibility. 245
7.3 Orthogonal Projection. 248
7.4 Idempotent Matrices. 250
7.5 Matrix Representation of Projectors. 256
CHAPTER 8. GENERALIZED INVERSES
8.1 Right and Left Inverses. 264
8.2 Generalized Inverse (y-inverse) . 265
8.3 Geometric Approach: LMN-inverse . 282
8.4 Minimum Norm Solution. 288
Contents xvii
8.5 Least Squares Solution. 289
8.6 Minimum Norm Least Squares Solution. 291
8.7 Various Types of «/-inverses. 292
8.8 G-inverses Through Matrix Approximations. 296
8.9 Gauss-Markov Theorem. 300
CHAPTER 9. MAJORIZATION
9.1 Majorization. 303
9.2 A Gallery of Functions. 307
9.3 Basic Results. 308
CHAPTER 10. INEQUALITIES FOR EIGENVALUES
10.1 Monotonicity Theorem. 322
10.2 Interlace Theorems . 328
10.3 Courant-Fischer Theorem. 332
10.4 Poincaré Separation Theorem. 337
10.5 Singular Values and Eigenvalues. 339
10.6 Products of Matrices, Singular Values, and Horn's
Theorem . 340
10.7 Von Neumann's Theorem. 342
CHAPTER 11. MATRIX APPROXIMATIONS
11.1 Norm on a Vector Space. 361
11.2 Norm on Spaces of Matrices.'. 363
11.3 Unitarily Invariant Norms . 374
11.4 Some Matrix Optimization Problems. 383
11.5 Matrix Approximations. 388
11.6 M, N-invariant Norm and Matrix Approximations. 394
11.7 Fitting a Hyperplane to a Set of Points. 398
CHAPTER 12. OPTIMIZATION PROBLEMS IN
STATISTICS AND ECONOMETRICS
12.1 Linear Models. 403
12.2 Some Useful Lemmas. 403
12.3 Estimation in a Linear Model. 406
12.4 A Trace Minimization Problem. 409
12.5 Estimation of Variance. 413
xviii MATRIX ALGEBRA THEORY AND APPLICATIONS
12.6 The Method of MINQUE: A Prologue. 415
12.7 Variance Components Models and Unbiased Estimation . 416
12.8 Normality Assumption and Invariant Estimators. 419
12.9 The Method of MINQUE. 422
12.10 Optimal Unbiased Estimation. 425
12.11 Total Least Squares. 428
CHAPTER 13. QUADRATIC SUBSPACES
13.1 Basic Ideas . 433
13.2 The Structure of Quadratic Subspaces. 438
13.3 Commutators of Quadratic Subspaces. 442
13.4 Estimation of Variance Components. 443
CHAPTER 14. INEQUALITIES WITH APPLICATIONS
IN STATISTICS
14.1 Some Results on nnd and pd Matrices. 449
14.2 Cauchy-Schwartz and Related Inequalities. 454
14.3 Hadamard Inequality. 456
14.4 Holder's Inequality. 457
14.5 Inequalities in Information Theory. 458
14.6 Convex Functions and Jensen's Inequality. 459
14.7 Inequalities Involving Moments. 461
14.8 Kantorovich Inequality and Extensions. 462
CHAPTER 15. NON-NEGATIVE MATRICES
15.1 Perron-ïïobenius Theorem. 467
15.2 Leontief Models in Economics. 477
15.3 Markov Chains. 481
15.4 Genetic Models. 485
15.5 Population Growth Models. 489
CHAPTER 16. MISCELLANEOUS COMPLEMENTS
16.1 Simultaneous Decomposition of Matrices. 493
16.2 More on Inequalities. 494
16.3 Miscellaneous Results on Matrices. 497
16.4 Toeplitz Matrices. 501
16.5 Restricted Eigenvalue Problem. 506
Contents xix
16.6 Product of Two Raleigh Quotients. 507
16.7 Matrix Orderings and Projection . 508
16.8 Soft Majorization. 509
16.9 Circulants. 511
16.10 Hadamard Matrices. 514
16.11 Miscellaneous Exercises. 515
REFERENCES. 519
INDEX. 529 |
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| spelling | Rao, Calyampudi Radhakrishna 1920-2023 Verfasser (DE-588)119285924 aut Matrix algebra and its applications to statistics and econometrics C. Radhakrishna Rao ; M. Bhaskara Rao Reprint. New Jersey[u.a.] World Scientific 2007 XIX, 535 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Matrizenalgebra (DE-588)4139347-8 gnd rswk-swf Matrizenalgebra (DE-588)4139347-8 s DE-604 Bhaskara Rao, M. Verfasser (DE-588)120849720 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015712816&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4139347-8 |
| title | Matrix algebra and its applications to statistics and econometrics |
| title_auth | Matrix algebra and its applications to statistics and econometrics |
| title_exact_search | Matrix algebra and its applications to statistics and econometrics |
| title_exact_search_txtP | Matrix algebra and its applications to statistics and econometrics |
| title_full | Matrix algebra and its applications to statistics and econometrics C. Radhakrishna Rao ; M. Bhaskara Rao |
| title_fullStr | Matrix algebra and its applications to statistics and econometrics C. Radhakrishna Rao ; M. Bhaskara Rao |
| title_full_unstemmed | Matrix algebra and its applications to statistics and econometrics C. Radhakrishna Rao ; M. Bhaskara Rao |
| title_short | Matrix algebra and its applications to statistics and econometrics |
| title_sort | matrix algebra and its applications to statistics and econometrics |
| topic | Matrizenalgebra (DE-588)4139347-8 gnd |
| topic_facet | Matrizenalgebra |
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