Representations of the group inverse of some 2 x 2 block matrices

New Representations of the Group Inverse of 2×2 Block Matrices

This paper presents a full rank factorization of a 2 × 2 block matrix without any restriction concerning the group inverse. Applying this factorization, we obtain an explicit representation of the group inverse in terms of four individual blocks of the partitioned matrix without certain restriction. We also derive some important coincidence theorems, including the expressions of the group inver...

A note on the representations for the Drazin inverse of 2 × 2 block matrices

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix M = [ A B C D ] in terms of its various blocks, where the blocks A and D are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of the Drazin inverse of M , denoted by MD , has recently been obtained under the assumptions th...

Representations and sign pattern of the group inverse for some block matrices

Let M= ( A B C 0 ) be a complex square matrix where A is square. When BCB = 0, rank(BC) = rank(B) and the group inverse of ( BAB 0 CB 0 ) exists, the group inverse of M exists if and only if rank(BC + A ( BAB ) π BA) = rank(B). In this case, a representation of M in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a...

Representations for the Drazin inverse of block cyclic matrices

Ela Representations for the Drazin Inverse of Block Cyclic Matrices

REPRESENTATIONS FOR THE DRAZIN INVERSE OF BLOCK CYCLIC MATRICES M. CATRAL AND P. VAN DEN DRIESSCHE Abstract. A formula for the Drazin inverse of a block k-cyclic (k ≥ 2) matrix A with nonzeros only in blocks Ai,i+1, for i = 1, . . . , k (mod k) is presented in terms of the Drazin inverse of a smaller order product of the nonzero blocks of A, namely Bi = Ai,i+1 · · ·Ai−1,i for some i. Bounds on ...