Some block matrices with signed Drazin inverses (Q445840)

The sign pattern of a real matrix \(A\) is the \(\left( 0,1,-1\right) - \)matrix obtained from \(A\) by replacing each entry by its sign, denoted by \(\operatorname{sgn}A\). A square real matrix \(A\) is said to be sign symmetric, if \(\operatorname{sgn}A\) is a symmetric matrix. \(A\) has signed Drazin inverse if \(\operatorname{sgn}\tilde{A}^{d}=\operatorname{sgn}A^{d}\) for each matrix \(\tilde{A}\in Q\left( A\right)\), where \(A^{d}\) denotes the Drazin inverse of \(A\) and \(Q\left( A\right) \) the set of real matrices with the same sign pattern as \(A\). The paper gives a complete characterization for: a) a class of anti-triangular matrices \( \left( \begin{matrix} A & B \\ C & 0 \end{matrix} \right) \) with signed Drazin inverse and b) for sign symmetric biparite matrices \(\left( \begin{matrix} 0 & B \\ C & 0 \end{matrix} \right) \) (where the zero blocks are square) with signed Drazin inverse.