Pseudo-Inverses in Associative Rings and Semigroups

Positivity of discrete singular systems and their stability: An LP-based approach

“…Hence, we first present some basic properties of this kind of inverse (see [8,13] for more details). For any matrix M ∈ R n×n there always exists a unique matrix M D , which is called the Drazin inverse of…”

Gröbner basis computation of Drazin inverses with multivariate rational function entries

“…Drazin in [6]) are used in many applications as singular differential equations and difference equations (see [4]), in finite finite Markov chains (see [16]), etc (see [1]). Many authors have addressed the problem of computing Drazin inverses where the matrix entries are indeed polynomials (see e.g.…”

Further results on the reverse order law for the group inverse in rings

“…for some nonnegative integer k. It can be proved (see [7,Theorem 1]) that such b is unique and it is customarily denoted a d . The least nonnegative integer k for which these equalities hold is the Drazin index i(a) of a.…”

“…The least nonnegative integer k for which these equalities hold is the Drazin index i(a) of a. In [7,Theorem 4] it was proved that an element a ∈ R is Drazin invertible if and only if there are nonnegative integers p, q and u, v ∈ R such that a p+1 u = a p and va q+1 = a q . The smallest value of p for which {u ∈ R : a p+1 u = a p } = ∅ is called the left index of a, denoted by l(a).…”

“…In a similar way the right index of a is defined, and is denoted by r(a). In a remark following [7,Theorem 4] it was shown that in case that a is Drazin invertible, then i(a) = l(a) = r(a).…”