Nashed Generalized Inverses and Applications

Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin-Madison, October 8 - 10, 1973
1. Auflage 2014
ISBN: 978-1-4832-7029-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 1068 Seiten, Web PDF

Generalized Inverses and Applications, contains the proceedings of an Advanced Seminar on Generalized Inverses and Applications held at the University of Wisconsin-Madison on October 8-10, 1973 under the auspices of the university's Mathematics Research Center. The seminar provided a forum for discussing the basic theory of generalized inverses and their applications to analysis and operator equations. Numerical analysis and approximation methods are considered, along with applications to statistics and econometrics, optimization, system theory, and operations research. Comprised of 14 chapters, this book begins by describing a unified approach to generalized inverses of linear operators, with particular reference to algebraic, topological, extremal, and proximinal properties. The reader is then introduced to the algebraic aspects of the generalized inverse of a rectangular matrix; the Fredholm pseudoinverse; and perturbations and approximations for generalized inverses and linear operator equations. Subsequent chapters deal with various applications of generalized inverses, including programming, games, and networks, as well as estimation and aggregation in econometrics. This monograph will be of interest to mathematicians and students of mathematics.

Autoren/Hrsg.

Nashed, M. Zuhair

Inhaltsverzeichnis

1;Front Cover;1
2;Generalized Inverses and Applications;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword;10
6;Preface;12
7;Chapter 1. A Unified Operator Theory of Generalized Inverses;16
7.1;ABSTRACT;16
7.2;Introduction;18
7.3;1. Algebraic Theory of Generalized Inverses;18
7.4;2. Generalized Inverses in Topological Vector Spaces;37
7.5;3. Extremal and Proximinal Properties of Generalized Inverses. Right - Orthogonal, Left - Orthogonal. Orthogonal. and Metric Generalized Inverses.;52
7.6;4. Specific Types of Algebraic Generalized Inverses;61
7.7;5. Generalized Inverses of Topological Homomorphisms in Topological Vector Spaces. Generalized Inverses in Banach and Hubert Spaces;67
7.8;6. Generalized Inverses of Adjoints and Operational Properties;84
7.9;7. Examples and Generalized Inverses for Special Classes of Operators;88
7.10;8. Generalized Inverses in Various Algebraic Structures;98
7.11;REFERENCES;117
8;Chapter 2. Algebraic Aspects of the Generalized Inverse of a Rectangular Matrix;126
8.1;1. Introduction;126
8.2;2. Definitions and uniqueness theorem;127
8.3;3. Existence theorems;128
8.4;4. Orthogonality and positivity;133
8.5;5. Conclusions;137
8.6;6. Acknowledgements;138
8.7;REFERENCES;138
9;Chapter 3. Some Topics in Generalized Inverses of Matrices;140
9.1;1. Introduction;140
9.2;2. Notation and terminology;140
9.3;3. Linear equations and {1}-inverses.;141
9.4;4. Projection and minimal properties of generalized inverses;144
9.5;5. Full-rank factorizations and partitioned matrices;149
9.6;6. Spectral generalized inverses;152
9.7;7. A spectral theory for rectangular matrices;154
9.8;8. Example; The orthogonal projector on the intersection of subspaces;158
9.9;References;159
10;Chapter 4. The Fredholm Pseudoinverse— An Analytic Episode in the History of Generalized Inverses;164
10.1;1. Historical background.;164
10.2;2. Pseudoinverses and the solution of equations.;165
10.3;3. Other properties of pseudoinverses.;167
10.4;4. The Fredholm resolvent.;169
10.5;5. The Fredholm pseudoinverse.;174
10.6;6. The Hurwitz pseudoresolvent.;180
10.7;7. An analytic pseudoinverse.;183
10.8;REFERENCES;187
11;Chapter 5. A Role of the Pseudoinverse in Analysis;190
11.1;1. Introduction;190
11.2;2. Operators;190
11.3;3. Units and Relative Self-Adjointness;193
11.4;4. A Spectral Theory for Operators;195
11.5;5. Examples;197
11.6;6. A Ternary Algebra.;203
11.7;7. Hubert Spaces Associated with Operators;205
11.8;REFERENCES;207
12;Chapter 6. Aspects of Generalized Inverses in Analysis and Regularization;208
12.1;I. ASPECTS OF GENERALIZED INVERSES IN LINEAR ANALYSIS;208
12.2;II. USES OF GENERALIZED INVERSES IN NONLINEAR ANALYSIS;229
12.3;III. ASPECTS OF GENERALIZED INVERSES IN REGULARIZATION AND APPROXIMATION OF ILL-POSED LINEAR OPERATOR EQUATIONS;240
13;Chapter 7. Methods for Computing the Moorse-Penrose Generalized Inverse, and Related Matters;260
13.1;1. Introduction;260
13.2;2. Mathematical preliminaries;263
13.3;3. Continuity of the generalized inverse, and the reduction of rank by perturbation;266
13.4;4. Perturbation theory;269
13.5;5. Special results for the full rank case;274
13.6;6. Methods based on Gauss elimination;277
13.7;7. Householder transformations;283
13.8;8. Modified Gram-Schmidt orthogonalization;289
13.9;9. Computing the generalized inverse of full-rank matrices;293
13.10;10. Computing the generalized inverse of an m X n matrix, rank k;300
13.11;11. Conclusions on computing methods;305
13.12;12. Concluding comments;307
13.13;Appendix A; Singular values and singular vectors;308
13.14;Appendix B; Additional Notes;311
13.15;REFERENCES;313
14;Chapter 8. Differentiation of Pseudoinverses, Separable Nonlinear Least Square Problems and Other Tales;318
14.1;1. Introduction;318
14.2;2. Preliminary Results;318
14.3;3. Differentiation of Projectors and Pseudoinverses;325
14.4;4. Variable Projections for Separable Nonlinear Least Squares;329
14.5;REFERENCES;336
15;Chapter 9. Perturbations and Approximations for Generalized Inverses and Linear Operator Equations;340
15.1;1. Introduction;342
15.2;2. Generalized Inverses of Linear Operators in Banach and Hilbert Spaces;344
15.3;3. Perturbations and Continuity of Generalized Inverses of Linear Operators;348
15.4;4. A New Approach to Approximations and Perturbation Bounds for Generalized Inverses. Collectively Compact Pointwise Convergent Approximations;367
15.5;5. Projection Methods for Generalized Inverses and Best Approximate Solutions of Linear Operator Equations of the First and Second Kinds;380
15.6;6. Iterative Methods for Generalized Inverses and Linear Operator Equations. Series and Integral Representations of Generalized Inverses;393
15.7;REFERENCES;403
16;Chapter 10. The Operator Pseudoinverse in Control and Systems Identification;412
16.1;PART 0 INTRODUCTION;413
16.2;PART I PSEUDOINVERSE OPERATORS IN HILBERT SPACE;417
16.3;PART II A GAUSS-MARKOV THEOREM FOR HILBERT SPACE;432
16.4;PART III AN APPLICATION TO SYSTEM IDENTIFICATION;445
16.5;PART IV ON THE QUADRATIC REGULATOR PROBLEM;460
16.6;PART V PSEUDOINVERSE OPERATOR APPROXIMATIONS;471
16.7;APPENDIX A. SOME PROPERTIES OF OPERATORS IN HILBERT SPACES;488
16.8;APPENDIX B HILBERT-SPACE-VALUED RANDOM VARIABLES;499
16.9;REFERENCES;506
17;Chapter 11. Applications of Generalized Inverses to Programming,Games and Networks;510
17.1;1. Introduction;510
17.2;2. The Bott-Duffin inverse and network analysis.;512
17.3;3. Explicit solutions of interval linear programs.;518
17.4;4. Integer and mixed integer solutions of linear equations.;524
17.5;5. Equilibrium points of bimatrix games.;527
17.6;REFERENCES;532
18;Chapter 12. Statistical Applications of the Pseudo Inverse;540
18.1;0. Introduction;540
18.2;I. Constrained Least Squares;541
18.3;II. The Singular Normal Density and Maximum Likelihood Estimates;543
18.4;III. Best Linear Unbiased Estimates; The Gauss-Markov Theorem.;544
18.5;IV. When is a Naive L. S. E. a B. L. U. E.?;546
18.6;V. The Distribution Theory for Quadratic Forms in Normal R. V. ' s .;547
18.7;VI. Sums of Squares.;549
18.8;VII. Conditional Expectations and Covariances;558
19;Chapter 13. Estimation and Aggregation in Econometrics: An Application of the Theory of Generalized Inverses;564
19.1;Introduction;565
19.2;PART 1. THEORY OF ESTIMATION;573
19.3;PART 2. THEORY OF AGGREGATION;633
19.4;REFERENCES;771
20;Chapter 14. Annotated Bibliography on Generalized Inverses and Applications;786
20.1;Introduction;786
20.2;Annotations;788
20.3;I. General;788
20.4;II. Theory of Generalized Inverse Matrices and Operators;791
20.5;III. Generalized Inverses in Analysis;819
20.6;IV. Numerical Analysis and Approximation Theory for Generalized Inverse Matrices and Operators, Least Squares Solutions. Computational Methods and Packages;836
20.7;V. Applications of Generalized Inverses;851
20.8;BIBLIOGRAPHY ON GENERALIZED INVERSES AND APPLICATIONS;866
21;Index;1058