Pseudo core inverses in rings with involution

Abstract

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary * -ring case by Rakić, Dinčić and Djordjević in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of * -ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.

Characterizations of *-DMP matrices over rings

“…Dually, the dual pseudo core inverse of A ∈ R n×n , denoted by A D ⃝ , is the unique solution to equations A m+1 X = A m for some positive integer m, X 2 A = X and (XA) * = XA [9]. The smallest positive integer m satisfying the above equations is called the dual pseudo core index of A , denoted by i(A) as well.…”

“…Also, the pseudo core inverse extended the core-EP inverse [13] from complex matrices to matrices over rings in terms of equations (see [9]). For consistency and convenience, we use the terminology 'pseudo core inverse' throughout this paper.…”

“…The pseudo core inverse of A ∈ R n×n , denoted by A The smallest positive integer m satisfying the above equations is called the pseudo core index of A . If A is pseudo core invertible, then it must be Drazin invertible, and the pseudo core index coincides with the Drazin index (see [9]). Thus, here and subsequently, we denote the pseudo core index of A by i…”

“…In [9], the authors introduced the notion of the pseudo core inverse, which extends the notion of core inverse to matrices of an arbitrary index. The pseudo core inverse of A ∈ R n×n , denoted by A The smallest positive integer m satisfying the above equations is called the pseudo core index of A .…”

Rank constrained matrix best approximation problem

“…In [16,20, 27,29], their algebraic properties and calculating methods are studied. In [9,22], the studying of them 20 is extended to some new fields, for example, ring and operator, etc. Moreover, the inverses are used to study partial orders in [1,28,30,31].Consider the following equation:(1.2)Let M ∈ C n×n with Ind(M ) = k, and b ∈ R M k .…”