Acute perturbation of the group inverse

On the perturbation and expression for the core inverse of linear operator in Hilbert space

“…This is the principal reason that we can find the characterization for the core inverse of the perturbed operator to have the simplest possible expression. How about the expressions for the core inverse under the null space-preserving perturbation, rankpreserving perturbation, more generally, the stable perturbation or acute perturbation [4,8,9,10]? We hope to solve these more and general cases in the future.…”

Further results on the perturbation estimations for the Drazin inverse

“…Concluding remarks. For the general case of A ∈ C n×n with ind(A) = k ≥ 1, and B is an acute perturbation of A with ind(B) = j ≥ 1, Wei [31] conjectured that ρ(BB D − AA D ) < 1 is the acute perturbation of the Drazin inverse, if and only if B satisfies condition (C s ) [5], i.e., R(B j ) ∩ N (A k ) = {0} and N (B j ) ∩ R(A k ) = {0}, which is equivalent to the rank condition [5], rank(B j ) = rank(A k ) = rank(A k B j A k ).…”

“…Motivated by the acute perturbation of the Moore-Penrose inverse [21,26], weighted Moore-Penrose inverse [17] and the group inverse [31], we present the acute perturbation for the Drazin inverse with respect to the spectral radius instead of the spectral norm, which extends the recent results on the group inverse by Wei in [31]. Definition 2.1.…”

“…This note is organized as follows. In Section 2, we introduce the definition of the acute perturbation of the Drazin inverse, which is equivalent to the condition (C 1 ) for matrices [6], and present several characterizations of acute perturbations based on the results from [6,31]. In Section 3 we present our new results on the spectral radius and the spectral norm for the difference of BB g − AA D via two examples.…”

Minimum property of condition numbers for the Drazin inverse and singular linear equations

“…The theory and applications of the Drazin inverse has been a substantial growth over the past few decades [1][2][3][4][5][6][7], which is useful in various applications, for example, applications in singular linear systems, Markov chains and iterative methods were found in the literature [8][9][10][11][12].…”