Previous |  Up |  Next


sign pattern matrix; signed Drazin inverse; strong sign nonsingular matrix
The sign pattern of a real matrix $A$, denoted by $\mathop {\rm sgn} A$, is the $(+,-,0)$-matrix obtained from $A$ by replacing each entry by its sign. Let $\mathcal {Q}(A)$ denote the set of all real matrices $B$ such that $\mathop {\rm sgn} B=\mathop {\rm sgn} A$. For a square real matrix $A$, the Drazin inverse of $A$ is the unique real matrix $X$ such that $A^{k+1}X=A^k$, $XAX=X$ and $AX=XA$, where $k$ is the Drazin index of $A$. We say that $A$ has signed Drazin inverse if $\mathop {\rm sgn} \widetilde {A}^{\rm d}=\mathop {\rm sgn} A^{\rm d}$ for any $\widetilde {A}\in \mathcal {Q}(A)$, where $A^{\rm d}$ denotes the Drazin inverse of $A$. In this paper, we give necessary conditions for some block triangular matrices to have signed Drazin inverse.
[1] Brualdi, R. A., Chavey, K. L., Shader, B. L.: Bipartite graphs and inverse sign patterns of strong sign-nonsingular matrices. J. Comb. Theory, Ser. B 62 (1994), 133-150. DOI 10.1006/jctb.1994.1059 | MR 1290635 | Zbl 0807.05053
[2] Brualdi, R. A., Ryser, H. J.: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39 Cambridge University Press, Cambridge (1991). MR 1130611 | Zbl 0746.05002
[3] Brualdi, R. A., Shader, B. L.: Matrices of Sign-Solvable Linear Systems. Cambridge Tracts in Mathematics 116 Cambridge University Press, Cambridge (1995). MR 1358133 | Zbl 0833.15002
[4] S. L. Campbell, C. D. Meyer, Jr.: Generalized Inverses of Linear Transformations. Surveys and Reference Works in Mathematics 4 Pitman Publishing, London (1979). MR 0533666 | Zbl 0417.15002
[5] Catral, M., Olesky, D. D., Driessche, P. van den: Graphical description of group inverses of certain bipartite matrices. Linear Algebra Appl. 432 (2010), 36-52. MR 2566457
[6] Eschenbach, C. A., Li, Z.: Potentially nilpotent sign pattern matrices. Linear Algebra Appl. 299 (1999), 81-99. MR 1723710 | Zbl 0941.15012
[7] Shader, B. L.: Least squares sign-solvability. SIAM J. Matrix Anal. Appl. 16 (1995), 1056-1073. DOI 10.1137/S0895479894272372 | MR 1351455 | Zbl 0837.05032
[8] Shao, J.-Y., He, J.-L., Shan, H.-Y.: Matrices with special patterns of signed generalized inverses. SIAM J. Matrix Anal. Appl. 24 (2003), 990-1002. DOI 10.1137/S0895479802401485 | MR 2003317
[9] Shao, J.-Y., Hu, Z.-X.: Characterizations of some classes of strong sign nonsingular digraphs. Discrete Appl. Math. 105 (2000), 159-172. DOI 10.1016/S0166-218X(00)00182-7 | MR 1780469 | Zbl 0965.05050
[10] Shao, J.-Y., Shan, H.-Y.: Matrices with signed generalized inverses. Linear Algebra Appl. 322 (2001), 105-127. DOI 10.1016/S0024-3795(00)00233-0 | MR 1804116 | Zbl 0967.15002
[11] Thomassen, C.: When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse. Linear Algebra Appl. 119 (1989), 27-34. DOI 10.1016/0024-3795(89)90066-9 | MR 1005232 | Zbl 0673.05067
[12] Zhou, J., Bu, C., Wei, Y.: Group inverse for block matrices and some related sign analysis. Linear and Multilinear Algebra 60 (2012), 669-681. DOI 10.1080/03081087.2011.625498 | MR 2929177 | Zbl 1246.15009
[13] Zhou, J., Bu, C., Wei, Y.: Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437 (2012), 1779-1792. MR 2946359 | Zbl 1259.15008
Partner of
EuDML logo