A class of outer generalized inverses

Drazin, M. P.

doi:10.1016/j.laa.2011.09.004

Cited by 143 publications

(148 citation statements)

References 12 publications

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“…Then Later, our motivation is the notion of a ( , )-inverse of an element in a semigroup, introduced by Drazin in [15]. Following the result from [9], the representation of outer inverses given in Proposition 1 investigates ( , )-inverses.…”

Section: Proposition 2 (Urquhart Formula) Let ∈ C 푚×푛 푟mentioning

confidence: 99%

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Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

et al. 2017

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Conditions for the existence and representations of {2}-, {1}-, and {1, 2}-inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

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Section: Proposition 2 (Urquhart Formula) Let ∈ C 푚×푛 푟mentioning

confidence: 99%

“…More precisely, representations of = is a full-rank factorization of and is invertible. It is worth mentioning that Drazin in [15] generalized the concept of the outer inverse with the prescribed range and null space by introducing the concept of a ( , )-inverse in a semigroup. In the matrix case, this concept can be defined as follows.…”

Section: Remarkmentioning

confidence: 99%

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

et al. 2017

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“…Mary showed in particular that a # , a D and a † are the inverses of a along a, a n and a * , respectively ([4, Theorem 11]). In [6], Drazin introduced (b, c)-inverse in a semigroup. It follows that (d, d)-inverse of a is an inverse of a along d (Mary's inverse).…”

Section: One-sided Inverse Along An Element In Semigroupsmentioning

confidence: 99%

Further results on the inverse along an element in semigroups and rings

Zhu

Chen

Patrício

2015

Linear and Multilinear Algebra

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In this paper, we introduce a new notion in a semigroup S as an extension of Mary's inverse. Let a, d ∈ S. An element a is called left (resp. right) invertibleAn existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) π -regularity and left (right) * -regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.

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“…But, in recent years, a number of papers was published considering the generalized inverses and associated partial orders in rings, see for example [14]. Furthermore, some new generalized inverses, such as core and dual core inverse (see [1]), (b, c)-inverse (see [4]) and an inverse along an element (see [19]), are introduced. For that reason there is a need of unified theory of partial orders based on generalized inverses in rings.…”

Section: Introductionmentioning

confidence: 99%

Partial orders in rings based on generalized inverses – unified theory

Rakić

Djordjević

2015

Linear Algebra and its Applications

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MSC: 06A06 15A09 16U99Keywords: Matrix partial ordering G-based order relation Minus partial order Star partial order Sharp partial order Core partial order Generalized inverse Moore-Penrose inverse Group inverse Core inverse Ring Dedekind finite ringThe unified theory for matrix partial orders based on generalized inverses has already been done by Mitra. We consider a special kind of ring R, which generalizes the ring of linear operators on finite dimensional vector space, and extend Mitra's approach to it. Thus, we find necessary and sufficient condition for a G-based relation to be a partial order on R. Also, some known results are generalized and some new results are proved.

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A class of outer generalized inverses

Cited by 143 publications

References 12 publications

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Further results on the inverse along an element in semigroups and rings

Partial orders in rings based on generalized inverses – unified theory

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