×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ben-Israel A, Generalized Inverses: Theory and Applications, 2. ed. (2003)
[2] DOI: 10.1080/03081080903410262 · Zbl 1219.15003
[3] DOI: 10.1017/CBO9780511574733
[4] DOI: 10.1016/j.amc.2009.04.054 · Zbl 1181.15003
[5] DOI: 10.1080/03081080903092243 · Zbl 1204.15010
[6] Bu C, Electron. J. Linear Algebra. 18 pp 117– (2009)
[7] DOI: 10.1016/j.amc.2008.05.145 · Zbl 1159.15003
[8] DOI: 10.1137/0131035 · Zbl 0341.34001
[9] Cao C, J. Natural Sci. Heilongjiang Univ. 18 pp 5– (2001)
[10] Cao C, Electron. J. Linear Algebra 18 pp 600– (2009)
[11] DOI: 10.1016/j.amc.2011.05.027 · Zbl 1222.15004
[12] Cao C, Int. Math. Forum. 31 pp 1511– (2006) · Zbl 1119.15002
[13] Catral M, Electron. J. Linear Algebra 17 pp 219– (2008)
[14] DOI: 10.1016/j.laa.2009.07.025 · Zbl 1184.15004
[15] DOI: 10.1007/BF02116008 · Zbl 0736.90021
[16] DOI: 10.1137/040606685 · Zbl 1100.15003
[17] Horn RA, Matrix Analysis (1985) · Zbl 0576.15001
[18] DOI: 10.2307/2295937
[19] DOI: 10.1016/j.amc.2010.05.084 · Zbl 1204.15014
[20] Samuelson PA, Foundations of Economic Analysis (1947)
[21] DOI: 10.1137/S0895479894272372 · Zbl 0837.05032
[22] DOI: 10.1137/S0895479802401485 · Zbl 1040.15006
[23] DOI: 10.1016/S0024-3795(00)00233-0 · Zbl 0967.15002
[24] DOI: 10.1081/STM-120002779 · Zbl 1005.60093
[25] DOI: 10.1016/j.laa.2004.08.021 · Zbl 1072.15007
[26] DOI: 10.1002/nla.663 · Zbl 1240.65120
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.