On block triangular matrices with signed Drazin inverse

Abstract

The sign pattern of a real matrix A, denoted by sgnA, is the (+, −, 0)-matrix obtained from A by replacing each entry by its sign. Let Q(A) denote the set of all real matrices B such that sgnB = sgnA. For a square real matrix A, the Drazin inverse of A is the unique real matrix X such that A k+1 X = A k, XAX = X and AX = XA, where k is the Drazin index of A. We say that A has signed Drazin inverse if \(\operatorname{sgn} {\tilde A^d} = \operatorname{sgn} {A^d}\) for any \(\tilde A \in Q(A)\), where A d denotes the Drazin inverse of A. In this paper, we give necessary conditions for some block triangular matrices to have signed Drazin inverse.