A note on the Drazin inverse of an anti-triangular matrix

Group Inverse and Generalized Drazin Inverse of Block Matrices in a Banach Algebra

“…Several authors have investigated representations for the group inverse and Drazin inverse of an anti-triangular block operator or matrix, under some conditions on the individual blocks [1,3,5,6,10].…”

The Drazin inverse of the sum of two matrices and its applications

“…This problem was first proposed by Campbell and Meyer [2], and is quite complicated. To the best of our knowledge, there was no explicit formula for the Drazin inverse of M. Some special cases have been considered, which can be found in [18,8,19,20,3,[21][22][23][24][25][26][27][28][29][30].…”

A note on the group inverse of some 2 × 2 block matrices over skew fields

“…Until now, this problem has not been solved completely. However, under some conditions, there have been some results about this problem (see [7][8][9][10][11][23][24][25]). In 1983, on the background of second-order systems of differential equations, Campbell in [4] proposed an open problem to give an explicit representation for the Drazin inverse of a 2 Â 2 antitriangular block matrix C A B 0 , where C is a square matrix and 0 is a square zero matrix.…”

“…For example, it can be applied to solve constrained optimization problems (KKT Linear Systems), and also can be used in finding the solution of differential equations (see [20-22]). About this open problem, there have been some results under some certain conditions (see [10][11][12][13][14][15][16][17][18][23][24][25]). Specially, in papers [15,16,18,24], the existence and the representations of the group inverse for the following block matrices are researched: [24]); [15]);…”