Geodesic
On block triangular matrices with signed Drazin inverse
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 883-892.

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The sign pattern of a real matrix $A$, denoted by $\mathop {\rm sgn} A$, is the $(+,-,0)$-matrix obtained from $A$ by replacing each entry by its sign. Let $\mathcal {Q}(A)$ denote the set of all real matrices $B$ such that $\mathop {\rm sgn} B=\mathop {\rm sgn} A$. 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 $\mathop {\rm sgn} \widetilde {A}^{\rm d}=\mathop {\rm sgn} A^{\rm d}$ for any $\widetilde {A}\in \mathcal {Q}(A)$, where $A^{\rm d}$ denotes the Drazin inverse of $A$. In this paper, we give necessary conditions for some block triangular matrices to have signed Drazin inverse.
DOI : 10.1007/s10587-014-0141-6
Classification : 15A09, 15B35
Keywords: sign pattern matrix; signed Drazin inverse; strong sign nonsingular matrix 
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Bu, Changjiang; Wang, Wenzhe; Zhou, Jiang; Sun, Lizhu. On block triangular matrices with signed Drazin inverse. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 883-892. doi : 10.1007/s10587-014-0141-6. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0141-6/

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