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Group inverse for block matrices and some related sign analysis. (English) Zbl 1246.15009

The sign pattern of a matrix \(M\) with real entries is a matrix with entries from \(\{0,1,-1\}\) that is obtained from \(M\) by replacing each entry by its sign. Let \(A \in \mathbb{R}^{n \times n}\) be group invertible (equivalently, there is a matrix \(X \in \mathbb{R}^{n \times n}\) that satisfies the equations \(AXA=A;XAX=X\) and \(AX=XA\); Such an \(X\) is called the (unique) group inverse of \(A\)). Let \(Q(A)\) denote the set of all real matrices with the same sign pattern as \(A\). For any \(B \in Q(A)\), if \(B\) is group invertible such that the group inverses of \(A\) and \(B\) have the same sign pattern, then \(A\) is called an \(S^2GI\) matrix. In the paper under review, the authors give sufficient conditions on certain block matrices to belong to the class \(S^2GI\). An application in determining sign patterns of solutions of systems of linear equations is studied.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15B35 Sign pattern matrices
15A06 Linear equations (linear algebraic aspects)
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