Screen LibraryMatrix algebra and its applications to statistics and econometrics /
Written by two top statisticians with experience in teaching matrix methods for applications in statistics, econometrics and related areas, this book provides a comprehensive treatment of the latest techniques in matrix algebra. A well-balanced approach to discussing the mathematical theory and appl...
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| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; Hackensack, N.J. : |
| Publication Dates: | 1998. |
| Literature type: | eBook |
| Language: | English |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/3599#t=toc |
| Summary: |
Written by two top statisticians with experience in teaching matrix methods for applications in statistics, econometrics and related areas, this book provides a comprehensive treatment of the latest techniques in matrix algebra. A well-balanced approach to discussing the mathematical theory and applications to problems in other areas is an attractive feature of the book. It can be used as a textbook in courses on matrix algebra for statisticians, econometricians and mathematicians as well. Some of the new developments of linear models are given in some detail using results of matrix algebra. |
| Carrier Form: | 1 online resource (xix,535pages) : illustrations |
| Bibliography: | Includes bibliographical references (pages 519-528) and index. |
| ISBN: | 9789812779281 (electronic bk.) |
| Index Number: | QA188 |
| CLC: | O241.6 |
| Contents: |
ch. 1. Vector spaces. 1.1. Rings and fields. 1.2. Mappings. 1.3. Vector spaces. 1.4. Linear independence and basis of a vector space. 1.5. Subspaces. 1.6. Linear equations. 1.7. Dual space. 1.8. Quotient space. 1.9. Projective geometry -- ch. 2. Unitary and Euclidean spaces. 2.1. Inner product. 2.2. Orthogonality. 2.3. Linear equations. 2.4. Linear functionals. 2.5. Semi-inner product. 2.6. Spectral theory. 2.7. Conjugate bilinear functionals and singular value decomposition -- ch. 3. Linear trasformations and matrices. 3.1. Preliminaries. 3.2. Algebra of transformations. 3.3. Inverse transformations. 3.4. Matrices -- ch. 4. Characteristics of matrices. 4.1. Rank and nullity of a matrix. 4.2. Rank and product of matrices. 4.3. Rank factorization and further results. 4.4. Determinants. 4.5. Determinants and minors -- ch. 5. Factorization of matrices. 5.1. Elementary matrices. 5.2. Reduction of general matrices. 5.3. Factorization of matrices with complex entries. 5.4. Eigenvalues and eigenvectors. 5.5. Simultaneous reduction of two matrices. 5.6. A review of matrix factorizations -- ch. 6. Operations on matrices. 6.1. Kronecker product. 6.2. The vec operation. 6.3. The Hadamard-Schur product. 6.4. Khatri-Rao product. 6.5. Matrix derivatives -- ch. 7. Projectors and idempotent operators. 7.1. Projectors. 7.2. Invariance and reducibility. 7.3. Orthogonal projection. 7.4. Idempotent matrices. 7.5. Matrix representation of projectors -- ch. 8. Generalized inverses. 8.1. Right and left inverses. 8.2. Generalized inverse (g-inverse). 8.3. Geometric approach: LMN-inverse. 8.4. Minimum norm solution. 8.5. Least squares solution. 8.6. Minimum norm least squares solution. 8.7. Various types of g-inverses. 8.8. G-inverses through matrix approximations. 8.9. Gauss-Markov theorem -- ch. 9. Majorization. 9.1. Majorization. 9.2. A gallery of functions. 9.3. Basic results -- ch. 10. Inequalities for eigenvalues. 10.1. Monotonicity theorem. 10.2. Interlace theorems. 10.3. Courant-Fischer theorem. 10.4. Poincar separation theorem. 10.5. Singular values and eigenvalues. 10.6. Products of matrices, singular values, and Horn's theorem. 10.7. Von Neumann's theorem. ch. 11. Matrix approximations. 11.1. Norm on a vector space. 11.2. Norm on spaces of matrices. 11.3. Unitarily invariant norms. 11.4. Some matrix optimization problems. 11.5. Matrix approximations. 11.6. M, N-invariant norm and matrix approximations. 11.7. Fitting a hyperplane to a set of points -- ch. 12. Optimization problems in statistics and econometrics. 12.1. Linear models. 12.2. Some useful lemmas. 12.3. Estimation in a linear model. 12.4. A trace minimization problem. 12.5. Estimation of variance. 12.6. The method of MINQUE: a prologue. 12.7. Variance components models and unbiased estimation. 12.8. Normality assumption and invariant estimators. 12.9. The method of MINQUE. 12.10. Optimal unbiased estimation. 12.11. Total least squares -- ch. 13. Quadratic subspaces. 13.1. Basic ideas.13.2. The structure of quadratic subspaces. 13.3. Commutators of quadratic subspaces. 13.4. Estimation of variance components -- ch. 14. Inequalities with applications in statistics. 14.1. Some results on nnd and pd matrices. 14.2. Cauchy-Schwartz and related inequalities. 14.3. Hadamard inequality. 14.4. Holder's inequality. 14.5. Inequalities in information theory. 14.6. Convex functions and Jensen's inequality. 14.7. Inequalities involving moments. 14.8. Kantorovich inequality and extensions -- ch. 15. Non-negative matrices. 15.1. Perron-Frobenius theorem. 15.2. Leontief models in economics. 15.3. Markov chains. 15.4. Genetic models. 15.5. Population growth models -- ch. 16. Miscellaneous complements. 16.1. Simultaneous decomposition of matrices. 16.2. More on inequalities. 16.3. Miscellaneous results on matrices. 16.4. Toeplitz matrices. 16.5. Restricted eigenvalue problem. 16.6. Product of two Raleigh quotients. 16.7. Matrix orderings and projection. 16.8. Soft majorization. 16.9. Circulants. 16.10. Hadamard matrices. 16.11. Miscellaneous exercises. |