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Algebraic Theory of Generalized Inverses

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Table of Contents
1 Algebraic Basic Knowledge 1
1.1 Complex Matrices 1
1.1.1 Some Decompositions of Complex Matrices 1
1.1.2 The Index of a Square Complex Matrix 5
1.1.3 Idempotents, Projections and EP Matrices 6
1.2 Definitions and Examples of Rings 7
1.2.1 Basic Concepts and Examples 7
1.2.2 Some Extensions of Rings 9
1.2.3 Idempotents, Units and Regular Elements 12
1.2.4 One-Sided Invertibility and Invertibility 14
1.3 Semigroups, Rings and Categories with Involution 16
1.3.1 Definitions and Examples 16
1.3.2 Proper Involutions 20
1.3.3 The Gelfand-Naimark Property 21
1.4 Regularity and -Regularity of Rings 21
1.4.1 Regularity and FP-Injectivity 21
1.4.2 Regularity 24
1.5 Invertibility of the Difference and the Sum of Idempotents 28
1.5.1 Invertibility of the Difference of Idempotents 28
1.5.2 Invertibility of the Sum of Idempotents 31
2 Moore-Penrose Inverses 35
2.1 Moore-Penrose Inverses of Complex Matrices 36
2.2 Characterizations of Moore-Penrose Inverses of Elements in Semigroups or Rings 38
2.2.1 Moore-Penrose Inverses of Elements in a Semigroup 39
2.2.2 Moore-Penrose Inverses of Elements in a Ring 42
2.2.3 Moore-Penrose Inverses of Matrices over a Ring 49
2.3 The Moore-Penrose Inverse of a Product 55
2.3.1 The Moore-Penrose Inverse of a Product paq 55
2.3.2 The Moore-Penrose Inverse of a Matrix Product 61
2.4 Moore-Penrose Inverses of Differences and Products of Projections 64
2.5 Jacobson’s Lemma for Moore-Penrose Inverses 72
2.5.1 Jacobson’s Lemma for Moore-Penrose Inverses in a (Generalized) GN Ring 72
2.5.2 Jacobson’s Lemma for Moore-Penrose Inverses in a Ring 74
2.6 The Moore-Penrose Inverse of a 2×2 Block Matrix 78
2.7 The Moore-Penrose Inverse of a Companion Matrix 85
2.8 The Moore-Penrose Inverse of a Sum of Morphisms 89
3 Group Inverses 97
3.1 Group Inverses of Complex Matrices 97
3.2 Characterizations of Group Inverses of Elements in Semigroups and Rings 98
3.3 The Group Inverse of a Product paq 102
3.4 The Group Inverse of a Sum of Morphisms 105
3.5 The Group Inverse of the Sum of Two Group Invertible Elements 109
3.6 The Group Inverse of the Product of Two Regular Elements 117
3.7 Group Inverses of Block Matrices 122
3.8 Group Inverses of Companion Matrices Over a Ring 128
3.9 EP Elements 134
4 Drazin Inverses 143
4.1 Drazin Inverses of Complex Matrices 143
4.2 Drazin Inverses of Elements in Semigroups and Rings 146
4.3 Drazin Invertibility in Two Semigroups of a Ring 150
4.4 Jacobson’s Lemma and Cline’s Formula for Drazin Inverses 156
4.5 Additive Properties of Drazin Inverses of Elements 161
4.6 Drazin Inverses of Products and Differences of Idempotents 165
4.7 Drazin Inverses of Matrices Over a Ring 170
4.8 The Drazin Inverse of a Sum of Morphisms 181
5 Core Inverses 209
5.1 Core Inverses of Complex Matrices 210
5.2 Core Inverses of Elements in Rings with Involution 211
5.2.1 Equivalent Definitions and Characterizations of Core Inverses 212
5.2.2 Relationship Between Group Invertibility and Core Invertibility 214
5.2.3 Characterizations of Core Inverses by Algebraic Equations 216
5.3 Characterizations of Core Invertibility by Special Elements 222
5.3.1 Characterizations of Core Invertibility by Hermitian Elements or Projections in a Ring 222
5.3.2 Characterizations of Core Invertibility for a Regular Element by Units in a Ring 227
5.4 The Core Inverse of the Sum of Two Core Invertible Elements 231
5.5 The Core Inverse of a Product paq 242
5.6 Core Inverses of Companion Matrices 247
5.7 The Core Inverse of a Sum of Morphisms 253
6 Pseudo Core Inverses 265
6.1 Core-EP Inverses of Complex Matrices 265
6.2 Pseudo Core Inverses of Elements in Rings with Involution 271
6.3 Additive and Multiplicative Properties 282
6.4 Pseudo Core Inverses of Jacobson Pairs 285
6.5 Pseudo Core Inverses of ab and ba 290
6.6 The Pseudo Core Inverse of a Sum of Morphisms 293
6.7 The Pseudo Core Inverse of a Product 302
6.8 The Pseudo Core Inverse of a Low Triangular Matrix 307
References 315
Index 321
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