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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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##### References:
  Ben-Israel A, Generalized Inverses: Theory and Applications, 2. ed. (2003)  DOI: 10.1080/03081080903410262 · Zbl 1219.15003  DOI: 10.1017/CBO9780511574733  DOI: 10.1016/j.amc.2009.04.054 · Zbl 1181.15003  DOI: 10.1080/03081080903092243 · Zbl 1204.15010  Bu C, Electron. J. Linear Algebra. 18 pp 117– (2009)  DOI: 10.1016/j.amc.2008.05.145 · Zbl 1159.15003  DOI: 10.1137/0131035 · Zbl 0341.34001  Cao C, J. Natural Sci. Heilongjiang Univ. 18 pp 5– (2001)  Cao C, Electron. J. Linear Algebra 18 pp 600– (2009)  DOI: 10.1016/j.amc.2011.05.027 · Zbl 1222.15004  Cao C, Int. Math. Forum. 31 pp 1511– (2006) · Zbl 1119.15002  Catral M, Electron. J. Linear Algebra 17 pp 219– (2008)  DOI: 10.1016/j.laa.2009.07.025 · Zbl 1184.15004  DOI: 10.1007/BF02116008 · Zbl 0736.90021  DOI: 10.1137/040606685 · Zbl 1100.15003  Horn RA, Matrix Analysis (1985) · Zbl 0576.15001  DOI: 10.2307/2295937  DOI: 10.1016/j.amc.2010.05.084 · Zbl 1204.15014  Samuelson PA, Foundations of Economic Analysis (1947)  DOI: 10.1137/S0895479894272372 · Zbl 0837.05032  DOI: 10.1137/S0895479802401485 · Zbl 1040.15006  DOI: 10.1016/S0024-3795(00)00233-0 · Zbl 0967.15002  DOI: 10.1081/STM-120002779 · Zbl 1005.60093  DOI: 10.1016/j.laa.2004.08.021 · Zbl 1072.15007  DOI: 10.1002/nla.663 · Zbl 1240.65120
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