# On block triangular matrices with signed Drazin inverse

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## Abstract

The sign pattern of a real matrix *A*, denoted by sgn*A*, 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 sgn*B* = sgn*A*. 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.

## Keywords

sign pattern matrix signed Drazin inverse strong sign nonsingular matrix## MSC 2010

15B35 15A09## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014