《Algebraic Theory of Generalized Inverses》以環、半群、範疇等代數結構中的Moore-Penrose逆、群逆、Drazin逆、核逆、偽核逆為主線，介紹了這幾類廣義逆的代數特性（包括代數方程刻畫、存在性準則、表達式等等），揭示了代數結構的性質和廣義逆的性質之間的內在聯繫。從矩陣分解入手，介紹矩陣廣義逆的基本性質，以此類比，延伸到環、半群中的元素以及範疇中的態射的相關廣義逆；從線性代數、抽象代數的一些基礎知識講起，一直到本領域最前沿的內容。
作者介紹
陳建龍//張小向
目錄
1 Algebraic Basic Knowledge 1.1 Complex Matrices 1.1.1 Some Decompositions of Complex Matrices 1.1.2 The Index of a Square Complex Matrix 1.1.3 Idempotents，Projections and EP Matrices 1.2 Definitions and Examples of Rings 1.2.1 Basic Concepts and Examples 1.2.2 Some Extensions of Rings 1.2.3 Idempotents， Units and Regular Elements 1.2.4 One-Sided Invertibility and Invertibility 1.3 Semigroups， Rings and Categories with Involution 1.3.1 Definitions and Examples 1.3.2 Proper Involutions 1.3.3 The Gelfand-Naimark Property 1.4 Regularity and Regularity of Rings 1.4.1 Regularity and FP-Injectivity 1.4.2 Regularity 1.5 Invertibility of the Difference and the Sum of Idempotents 1.5.1 Invertibility of the Difference of Idempotents 1.5.2 Invertibility of the Sum of Idempotents 2 Moore-Penrose Inverses 2.1 Moore-Penrose Inverses of Complex Matrices 2.2 Characterizations of Moore-Penrose Inverses of Elements in Semigroups or Rings 2.2.1 Moore-Penrose Inverses of Elements in a Semigroup 2.2.2 Moore-Penrose Inverses of Elements in a Ring 2.2.3 Moore-Penrose Inverses of Matrices over a Ring 2.3 The Moore-Penrose Inverse of a Product 2.3.1 The Moore-Penrose Inverse of a Product paq 2.3.2 The Moore-Penrose Inverse of a Matrix Product 2.4 Moore-Penrose Inverses of Differences and Products of Projections 2.5 Jacobson』s Lemma for Moore-Penrose Inverses 2.5.1 Jacobson』s Lemma for Moore-Penrose Inverses in a (Generalized) GN Ring 2.5.2 Jacobson』s Lemma for Moore-Penrose Inverses in a Ring 2.6 The Moore-Penrose Inverse of a 2×2 Block Matrix 2.7 The Moore-Penrose Inverse of a Companion Matrix 2.8 The Moore-Penrose Inverse of a Sum of Morphisms 3 Group Inverses 3.1 Group Inverses of Complex Matrices 3.2 Characterizations of Group Inverses of Elements in Semigroups and Rings 3.3 The Group Inverse of a Product paq 3.4 The Group Inverse of a Sum of Morphisms 3.5 The Group Inverse of the Sum of Two Group Invertible Elements 3.6 The Group Inverse of the Product of Two Regular Elements 3.7 Group Inverses of Block Matrices 3.8 Group Inverses of Companion Matrices Over a Ring 3.9 EP Elements 4 Drazin Inverses 4.1 Drazin Inverses of Complex Matrices 4.2 Drazin Inverses of Elements in Semigroups and Rings 4.3 Drazin Invertibility in Two Semigroups of a Ring
4.4 Jacobson』s Lemma and Cline』s Formula for Drazin Inverses 4.5 Additive Properties of Drazin Inverses of Elements 4.6 Drazin Inverses of Products and Differences of Idempotents 4.7 Drazin Inverses of Matrices Over a Ring 4.8 The Drazin Inverse of a Sum of Morphisms 5 Core Inverses 5.1 Core Inverses of Complex Matrices 5.2 Core Inverses of Elements in Rings with Involution 5.2.1 Equivalent Definitions and Characterizations of Core Inverses 5.2.2 Relationship Between Group Invertibility and Core Invertibility 5.2.3 Characterizations of Core Inverses by Algebraic Equations 5.3 Characterizations of Core Invertibility by Special Elements 5.3.1 Characterizations of Core Invertibility by Hermitian Elements or Projections in a Ring 5.3.2 Characterizations of Core Invertibility for a Regular Element by Units in a Ring 5.4 The Core Inverse of the Sum of Two Core Invertible Elements 5.5 The Core Inverse of a Product paq 5.6 Core Inverses of Companion Matrices 5.7 The Core Inverse of a Sum of Morphisms 6 Pseudo Core Inverses 6.1 Core-EP Inverses of Complex Matrices 6.2 Pseudo Core Inverses of Elements in Rings with Involution 6.3 Additive and Multiplicative Properties 6.4 Pseudo Core Inverses of Jacobson Pairs 6.5 Pseudo Core Inverses of ab and ba 6.6 The Pseudo Core Inverse of a Sum of Morphisms 6.7 The Pseudo Core Inverse of a Product 6.8 The Pseudo Core Inverse of a Low Triangular Matrix References Index
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