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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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