| Preface | p. vii |
| Acknowledgements | p. ix |
| Glossary of Symbols and Abbreviations | p. xi |
| Introduction | p. 1 |
| Matrix orders | p. 1 |
| Parallel sum and shorted operator | p. 3 |
| A tour through the rest of the monograph | p. 4 |
| Matrix Decompositions and Generalized Inverses | p. 9 |
| Introduction | p. 9 |
| Matrix decompositions | p. 10 |
| Generalized inverse of a matrix | p. 17 |
| The group inverse | p. 26 |
| Moore-Penrose inverse | p. 36 |
| Generalized inverses of modified matrices | p. 46 |
| Simultaneous diagonalization | p. 55 |
| Exercises | p. 64 |
| The Minus Order | p. 67 |
| Introduction | p. 67 |
| Space pre-order | p. 68 |
| Minus order - Some characterizations | p. 72 |
| Matrices above/below a given matrix under the minus order | p. 81 |
| Subclass of g-inverses A- of A such that A- A = A-B and AA- = BA- when A <- B | p. 84 |
| Minus order for idempotent matrices | p. 93 |
| Minus order for complex matrices | p. 95 |
| Exercises | p. 98 |
| The Sharp Order | p. 103 |
| Introduction | p. 103 |
| Sharp order - Characteristic properties | p. 104 |
| Sharp order - Other properties | p. 110 |
| Drazin order and an extension | p. 117 |
| Exercises | p. 124 |
| The Star Order | p. 127 |
| Introduction | p. 127 |
| Star order - Characteristic properties | p. 128 |
| Subclasses of g-inverses for which A <* B | p. 136 |
| Star order for special subclasses of matrices | p. 138 |
| Star order and idempotent matrices | p. 145 |
| Fisher-Cochran type theorems | p. 150 |
| Exercises | p. 152 |
| One-Sided Orders | p. 155 |
| Introduction | p. 155 |
| The condition AA- = BA- | p. 156 |
| One-sided sharp order | p. 160 |
| Roles of A-c and A-a in one-sided sharp order | p. 167 |
| One-sided star order | p. 171 |
| Exercises | p. 180 |
| Unified Theory of Matrix Partial Orders through Generalized Inverses | p. 183 |
| Introduction | p. 183 |
| g-based order relations: Definitions and preliminaries | p. 184 |
| o-based order relations and their properties | p. 195 |
| One-sided g-based order relations | p. 200 |
| Properties of g-based order relations | p. 203 |
| On g-based extensions | p. 208 |
| Exercises | p. 212 |
| The Löwner Order | p. 215 |
| Introduction | p. 215 |
| Definition and basic properties | p. 215 |
| Löwner order on powers and its relation with other partial orders | p. 226 |
| Löwner order on generalized inverses | p. 230 |
| Generalizations of the Löwner order | p. 238 |
| Exercises | p. 243 |
| Parallel Sums | p. 245 |
| Introduction | p. 245 |
| Definition and properties | p. 246 |
| Parallel sums and partial orders | p. 259 |
| Continuity and index of parallel sums | p. 264 |
| Exercises | p. 270 |
| Schur Complements and Shorted Operators | p. 273 |
| Introduction | p. 273 |
| Shorted operator - A motivation | p. 274 |
| Generalized Schur complement and shorted operator | p. 276 |
| Shorted operator via parallel sums | p. 283 |
| Generalized Schur complement and shorted operator of a matrix over general field | p. 285 |
| Exercises | p. 293 |
| Shorted Operators - Other Approaches | p. 295 |
| Introduction | p. 295 |
| Shorted operator as the limit of parallel sums - General matrices | p. 296 |
| Rank minimization problem and shorted operator | p. 305 |
| Computation of shorted operator | p. 310 |
| Exercises | p. 315 |
| Lattice Properties of Partial Orders | p. 317 |
| Introduction | p. 317 |
| Supremum and infimum of a pair of matrices under the minus order | p. 318 |
| Supremum and infimum under the star order | p. 330 |
| Infimum under the sharp order | p. 338 |
| Exercises | p. 342 |
| Partial Orders of Modified Matrices | p. 343 |
| Introduction | p. 343 |
| Space pre-order | p. 344 |
| Minus order | p. 352 |
| Sharp order | p. 357 |
| Star order | p. 364 |
| Löwner order | p. 367 |
| Equivalence Relations on Generalized and Outer Inverses | p. 371 |
| Introduction | p. 371 |
| Equivalence relation on g-inverses of a matrix | p. 372 |
| Equivalence relations on subclasses of g-inverses | p. 380 |
| Equivalence relation on the outer inverses of a matrix | p. 384 |
| Diagrammatic representation of the g-inverses and outer inverses | p. 390 |
| The Ladder | p. 401 |
| Applications | p. 407 |
| Introduction | p. 407 |
| Point estimation in a general linear model | p. 407 |
| Comparison of models when model matrices are related under matrix partial orders | p. 411 |
| Shorted operators - Applications | p. 415 |
| Application of parallel sum and shorted operator to testing in linear models | p. 418 |
| Shorted operator adjustment for modification of network or mechanism | p. 418 |
| Some Open Problems | p. 423 |
| Simultaneous diagonalization | p. 423 |
| Matrices below a given matrix under sharp order | p. 424 |
| Partial order combining the minus and sharp orders | p. 424 |
| When is a g-based order relation a partial order? | p. 425 |
| Parallel sum and g-inverses | p. 425 |
| Shorted operator and a maximization problem | p. 426 |
| The ladder problem | p. 427 |
| Relations and Partial Orders | p. 429 |
| Introduction | p. 429 |
| Relations | p. 429 |
| Semi-groups and groups | p. 432 |
| Semi-groups and partial order | p. 433 |
| Involution | p. 435 |
| Compatibility of partial orders with algebraic operations | p. 435 |
| Partial orders induced by convex cones | p. 436 |
| Creating new partial orders from old partial orders | p. 436 |
| Bibliography | p. 439 |
| Index | p. 445 |
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