Representations for the group inverse of anti-triangular block operator matrices

Formulas for the Drazin inverse of matrices over skew fields

“…In 2006, the above result was extended to the case of AA π B = 0, CA π B = 0, S = 0 (see [18]). And there are some results on the representations for the Drazin (group) inverse of M (see [9], [12], [19]- [23], [28]). Here we list some cases:…”

General exact solutions of certain second-order homogeneous algebraic differential equations

“…It is worth noting that ever since 1980s, [2,3] it has been an open problem to find an explicit formula for the Drazin inverse of a general anti-triangular (operator) matrix. Although much progress has been made, [4][5][6][7][8][9][10][11][12] less has been done in the case that the Drazin index of the anti-triangular matrix turns out to be two, which is the concern of this paper.…”

“…In the case that X is a Hilbert space, M, C and K are positive semi-definite operators with the summation M + C + K being positive definite, and both M and 2M + C have closed ranges, we have managed to provide a formula for E D λ based on a new method initiated in [12]; see Theorem 3.10 and Corollary 3.11 below, which to the best of our knowledge are new even for matrices. It is worth noting that our framework restricted as in (3.1a) and (3.1b) covers especially a lot of physical systems arising in the area of structural dynamics, where M is real, symmetric and positive definite, while C and K are real, symmetric and positive definite or positive semi-definite.…”

Group invertible block matrices

“…is the generalized Schur complement of M. In addition, there are many results on the representations for group (Drazin) inverse of block matrices (operators) and their applications [10,11,17,19,22].…”