A class of outer generalized inverses

By M. Drazin. 2012

184 citations recorded by the source corpus.

Abstract In any * -semigroup or semigroup S , it is shown that the Moore–Penrose inverse y = a † , the author’s pseudo-inverse y = a ′ , Chipman’s weighted inverse and the Bott–Duffin inverse are all special cases of the more general class of “ ( b , c ) -inverses” y ∈ S satisfying y ∈ ( bSy ) ∩ ( ySc ) , yab = b and cay = c . These ( b , c ) -inverses always satisfy yay = y , are always unique when they exist, and exist if and only if b ∈ Scab and c ∈ cabS , in which case, under the partial order M of Mitsch, y is also the unique M -greatest element of the set X a = X a , b , c = { x : x ∈ S , xax = x and x ∈ ( bSx ) ∩ ( xSc ) } and the unique M -least element of Z a = Z a , b , c = { z : z ∈ S , zaz = z , zab = b and caz = c } . The above all holds in arbitrary semigroups S , hence in particular in any associative ring R . For any complex n × n matrices a , b , c , an efficient uniform procedure is given to compute the ( b , c ) -inverse of a whenever it exists. In the ring case, a ∈ R is called “weakly invertible” if there exist b , c ∈ R satisfying 1 - b ∈ ( 1 - a ) R , 1 - c ∈ R ( 1 - a ) such that a has a ( b , c ) -inverse y satisfying ay = ya , and it is shown that a is weakly invertible if and only if a is strongly clean in the sense of Nicholson, i.e. a = u + e for some unit u and idempotent e with eu = ue .

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/61d09f272aa8970d786bc8c2a7169a10c9caca64

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Citations in the corpus

Commuting properties of generalized inverses (2013, background): “This y is always unique and satisfies yay = y whenever it exists and, by appropriate choices of b and c (see [4], p.”
ON ONE-SIDED (B;C)-INVERSES OF ARBITRARY MATRICES (2017, background): “Recently, two important outer inverses have been introduced: the inverse along an element (see [15]) and the (b, c)-inverse (see [9]).”
Pseudo core inverses in rings with involution (2016, background): “hat, let us mention two known deﬁnitions. For a,b ∈ R, a ≤H b ⇔ Ra ⊆ Rb and aR ⊆ bR [13]. Deﬁnition 3.1. [13] Let a,d,x ∈ R. x is an inverse of a along d if xad = d = dax and x ≤H d. 9 Deﬁnition 3.2. [5] Let a,b,c,x ∈ R, x is said to be a (b,c)-inverse of a if x ∈ bRx∩xRc and xab = b, cax = c. In wh”
Group, Moore–Penrose, core and dual core inverse in rings with involution☆ (2014, background): “Drazin proved in [7] that a# is (a, a)-inverse of a and that a† is (a∗, a∗) inverse of a.”