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On block triangular matrices with signed Drazin inverse. (English) Zbl 1349.15084
For a real matrix \(A \in \mathbb{R}^{n \times n}\), the sign pattern matrix relative to \(A\) denoted by \(\mathrm{sgn}(A)\) is the matrix whose entries are \(+\), \(-\) and \(0\) obtained from \(A\) by replacing each of its entry by its sign. Let \(\mathcal{Q}(A)\) denote the set of all real matrices \(B\) such that \(\mathrm{sgn}(B)=\mathrm{sgn}(A)\). Let us recall that the index of a square matrix \(A\) is the smallest nonnegative integer \(k\) for which the ranks of \(A^k\) and \(A^{k+1}\) are the same. By a finite dimensionality argument, it follows that the index always exists. Now, let \(k\) be the index of \(A\). It is well known that there is a unique matrix \(X\) such that \(A^{k+1}X=A^k\), \(XAX=X\) and \(AX=XA\). Such an \(X\) is unique, is called the Drazin inverse of \(A\) and is denoted by \(A^D\). We say that \(A\) has a signed Drazin inverse if \(\mathrm{sgn}(C^D)=\mathrm{sgn}(A^D)\) for any \(C \in \mathcal{Q}(A)\). Finally, given \(A\), we say that \(\mathrm{sgn}(A)\) is potentially nilpotent if there exists a nilpotent matrix \(C \in \mathcal{Q}(A)\). In the work under review, the authors present characterizations for certain block triangular matrices to have signed Drazin inverses. Let us only provide a sample result: Let \(M\) be an upper triangular block square matrix which has a signed Drazin inverse and whose diagonal blocks are \(A\) and \(C\), where \(C\) is potentially nilpotent. Then, \(M\) is permutationally similar to a block square matrix whose upper triangular part is given by a matrix \(F\) and whose bottom right block is permutationally similar to \(C\) and is strictly upper triangular. Among other things, the authors show that \(\mathrm{sgn}((C^D)^2x^1) = \mathrm{sgn} ((A^D)^2f^1)\), for all \(C \in \mathcal{Q}(A)\) and \(X \in \mathcal{Q}(F)\), where \(x^1\) and \(f^1\) denote the first columns of \(X\) and \(F\), respectively.

MSC:
15B35 Sign pattern matrices
15A09 Theory of matrix inversion and generalized inverses
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