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El. knyga: Matrix Partial Orders, Shorted Operators and Applications [World Scientific e-book]

(Univ Of Hyderabad, India), (Hindu College, Delhi Univ, India), (Indian Statistical Inst, India)
  • Formatas: 464 pages
  • Serija: Series In Algebra 10
  • Išleidimo metai: 30-Jun-2009
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • ISBN-13: 9789812838452
Kitos knygos pagal šią temą:
  • World Scientific e-book
  • Kaina: 142,30 €*
  • * this price gives unlimited concurrent access for unlimited time
Matrix Partial Orders, Shorted Operators and ApplicationsDidesnis paveikslėlis
  • Formatas: 464 pages
  • Serija: Series In Algebra 10
  • Išleidimo metai: 30-Jun-2009
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • ISBN-13: 9789812838452
Kitos knygos pagal šią temą:
World Scientific e-booksMore info about World Scientific e-books
Partnerių interneto svetainė: http://www.worldscientific.com/worldscibooks/10.1142/7170#t=toc
The present monograph on matrix partial orders. the first on this topic. makes a unique presentation of many partial orders on matrices that have fascinated mathematicians for their beauty and applied scientists for their wide-ranging application potential. Except for the Lowner order. the partial orders considered are relatively new and came into being in the late 1970s. After a detailed introduction to generalized inverses and decompositions. the three basic partial orders---namely. the minus. the sharp and the star---and the corresponding one-sided orders are presented using various generalized inverses. The authors then give a unified theory of all these partial orders as well as study the parallel sums and shorted matrices. the latter being studied at great length. Partial orders of modified matrices are a new addition. Finally. applications are given in statistics and electrical network theory.

Preface vii
Acknowledgements ix
Glossary of Symbols and Abbreviations xi
1 Introduction
1(8)
1.1 Matrix orders
1(2)
1.2 Parallel sum and shorted operator
3(1)
1.3 A tour through the rest of the monograph
4(5)
2 Matrix Decompositions and Generalized Inverses
9(58)
2.1 Introduction
9(1)
2.2 Matrix decompositions
10(7)
2.3 Generalized inverse of a matrix
17(9)
2.4 The group inverse
26(10)
2.5 Moore-Penrose inverse
36(10)
2.6 Generalized inverses of modified matrices
46(9)
2.7 Simultaneous diagonalization
55(9)
2.8 Exercises
64(3)
3 The Minus Order
67(36)
3.1 Introduction
67(1)
3.2 Space pre-order
68(4)
3.3 Minus order-Some characterizations
72(9)
3.4 Matrices above/below a given matrix under the minus order
81(3)
3.5 Subclass of g-inverses A- of A such that A-A = A-B and AA- = BA- when A <-B
84(9)
3.6 Minus order for idempotent matrices
93(2)
3.7 Minus order for complex matrices
95(3)
3.8 Exercises
98(5)
4 The Sharp Order
103(24)
4.1 Introduction
103(1)
4.2 Sharp order-Characteristic properties
104(6)
4.3 Sharp order-Other properties
110(7)
4.4 Drazin order and an extension
117(7)
4.5 Exercises
124(3)
5 The Star Order
127(28)
5.1 Introduction
127(1)
5.2 Star order-Characteristic properties
128(8)
5.3 Subclasses of g-inverses for which A < B
136(2)
5.4 Star order for special subclasses of matrices
138(7)
5.5 Star order and idempotent matrices
145(5)
5.6 Fisher-Cochran type theorems
150(2)
5.7 Exercises
152(3)
6 One-Sided Orders
155(28)
6.1 Introduction
155(1)
6.2 The condition AA- = BA-
156(4)
6.3 One-sided sharp order
160(7)
6.4 Roles of A-c and A-a in one-sided sharp order
167(4)
6.5 One-sided star order
171(9)
6.6 Exercises
180(3)
7 Unified Theory of Matrix Partial Orders through Generalized Inverses
183(32)
7.1 Introduction
183(1)
7.2 G-based order relations: Definitions and preliminaries
184(11)
7.3 O-based order relations and their properties
195(5)
7.4 One-sided G-based order relations
200(3)
7.5 Properties of G-based order relations
203(5)
7.6 On G-based extensions
208(4)
7.7 Exercises
212(3)
8 The Lowner Order
215(30)
8.1 Introduction
215(1)
8.2 Definition and basic properties
215(11)
8.3 Lowner order on powers and its relation with other partial orders
226(4)
8.4 Lowner order on generalized inverses
230(8)
8.5 Generalizations of the Lowner order
238(5)
8.6 Exercises
243(2)
9 Parallel Sums
245(28)
9.1 Introduction
245(1)
9.2 Definition and properties
246(13)
9.3 Parallel sums and partial orders
259(5)
9.4 Continuity and index of parallel sums
264(6)
9.5 Exercises
270(3)
10 Schur Complements and Shorted Operators
273(22)
10.1 Introduction
273(1)
10.2 Shorted operator-A motivation
274(2)
10.3 Generalized Schur complement and shorted operator
276(7)
10.4 Shorted operator via parallel sums
283(2)
10.5 Generalized Schur complement and shorted operator of a matrix over general field
285(8)
10.6 Exercises
293(2)
11 Shorted Operators-Other Approaches
295(22)
11.1 Introduction
295(1)
11.2 Shorted operator as the limit of parallel sums-General matrices
296(9)
11.3 Rank minimization problem and shorted operator
305(5)
11.4 Computation of shorted operator
310(5)
11.5 Exercises
315(2)
12 Lattice Properties of Partial Orders
317(26)
12.1 Introduction
317(1)
12.2 Supremum and infimum of a pair of matrices under the minus order
318(12)
12.3 Supremum and infimum under the star order
330(8)
12.4 Infimum under the sharp order
338(4)
12.5 Exercises
342(1)
13 Partial Orders of Modified Matrices
343(28)
13.1 Introduction
343(1)
13.2 Space pre-order
344(8)
13.3 Minus order
352(5)
13.4 Sharp order
357(7)
13.5 Star order
364(3)
13.6 Lowner order
367(4)
14 Equivalence Relations on Generalized and Outer Inverses
371(36)
14.1 Introduction
371(1)
14.2 Equivalence relation on g-inverses of a matrix
372(8)
14.3 Equivalence relations on subclasses of g-inverses
380(4)
14.4 Equivalence relation on the outer inverses of a matrix
384(6)
14.5 Diagrammatic representation of the g-inverses and outer inverses
390(11)
14.6 The Ladder
401(6)
15 Applications
407(16)
15.1 Introduction
407(1)
15.2 Point estimation in a general linear model
407(4)
15.3 Comparison of models when model matrices are related under matrix partial orders
411(4)
15.4 Shorted operators-Applications
415(3)
15.5 Application of parallel sum and shorted operator to testing in linear models
418(1)
15.6 Shorted operator adjustment for modification of network or mechanism
418(5)
16 Some Open Problems
423(6)
16.1 Simultaneous diagonalization
423(1)
16.2 Matrices below a given matrix under sharp order
424(1)
16.3 Partial order combining the minus and sharp orders
424(1)
16.4 When is a G-based order relation a partial order?
425(1)
16.5 Parallel sum and g-inverses
425(1)
16.6 Shorted operator and a maximization problem
426(1)
16.7 The ladder problem
427(2)
Appendix A Relations and Partial Orders
429(10)
A.1 Introduction
429(1)
A.2 Relations
429(3)
A.3 Semi-groups and groups
432(1)
A.4 Semi-groups and partial orders
433(2)
A.5 Involution
435(1)
A.6 Compatibility of partial orders with algebraic operations
435(1)
A.7 Partial orders induced by convex cones
436(1)
A.8 Creating new partial orders from old partial orders
436(3)
Bibliography 439(6)
Index 445
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