Generalized Inverses

2021 ◽  
Vol 22 ◽  
pp. 13-32
Author(s):  
Dragan S. Djordjevic
Keyword(s):  
Generalized Inverses ◽  
Drazin Inverse ◽  
Reverse Order ◽  
Survey Paper ◽  
Reverse Order Law

In this survey paper we present some aspects of generalized inverses, which are related to inner and outer invertibility, Moore-Penrose inverse, the appropriate reverse order law, and Drazin inverse.

scholarly journals Some additive and multiplicative results for generalized inverses

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 2049-2057
Author(s):  
Jovana Nikolov-Radenkovic
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Reverse Order Law ◽  
Necessary And Sufficient ◽  
Regular Operators

In this paper we give necessary and sufficient conditions for A1{1,3} + A2{1, 3}+ ... + Ak{1,3} ? (A1 + A2 + ... + Ak){1,3} and A1{1,4} + A2{1,4} + ... + Ak{1,4} ? (A1 + A2 + ... + Ak){1,4} for regular operators on Hilbert space. We also consider similar inclusions for {1,2,3}- and {1,2,4}-i inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.

Central Drazin inverses

2019 ◽  
Vol 18 (04) ◽  
pp. 1950065 ◽  
Author(s):  
Cang Wu ◽  
Liang Zhao
Keyword(s):  
Drazin Inverse ◽  
Reverse Order ◽  
Basic Properties ◽  
Reverse Order Law ◽  
Invertible Elements

We introduce and study a subclass of the Drazin invertible elements in a ring [Formula: see text], which are called central Drazin invertible. An element [Formula: see text] is said to be central Drazin invertible if there exists [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for some integer [Formula: see text]. Some basic properties of the central Drazin inverse are obtained. Of particular interest are the central Drazin invertible elements that are simultaneously group invertible, which we show have a property generalizing strong cleanness. Some well-known results related to the cleanness of rings and the reverse order law are generalized.

scholarly journals The Expression of the Drazin Inverse with Rank Constraints

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Linlin Zhao
Keyword(s):  
Matrix Decomposition ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Reverse Order ◽  
Reverse Order Law ◽  
The Matrix ◽  
Rank Constraints

By using the matrix decomposition and the reverse order law, we provide some new expressions of the Drazin inverse for any2×2block matrix with rank constraints.

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