# zbMATH — the first resource for mathematics

Some block matrices with signed Drazin inverses. (English) Zbl 1259.15008
The sign pattern of a real matrix $$A$$ is the $$\left( 0,1,-1\right) -$$matrix obtained from $$A$$ by replacing each entry by its sign, denoted by $$\operatorname{sgn}A$$. A square real matrix $$A$$ is said to be sign symmetric, if $$\operatorname{sgn}A$$ is a symmetric matrix. $$A$$ has signed Drazin inverse if $$\operatorname{sgn}\tilde{A}^{d}=\operatorname{sgn}A^{d}$$ for each matrix $$\tilde{A}\in Q\left( A\right)$$, where $$A^{d}$$ denotes the Drazin inverse of $$A$$ and $$Q\left( A\right)$$ the set of real matrices with the same sign pattern as $$A$$. The paper gives a complete characterization for: a) a class of anti-triangular matrices $$\left( \begin{matrix} A & B \\ C & 0 \end{matrix} \right)$$ with signed Drazin inverse and b) for sign symmetric biparite matrices $$\left( \begin{matrix} 0 & B \\ C & 0 \end{matrix} \right)$$ (where the zero blocks are square) with signed Drazin inverse.

##### MSC:
 15A09 Theory of matrix inversion and generalized inverses 15B35 Sign pattern matrices
Full Text:
##### References:
  Brualdi, R.A.; Shader, B.L., Matrices of sign-solvable linear systems, (1995), Cambridge University Press Cambridge · Zbl 0833.15002  Bu, C.; Zhang, K.; Zhao, J., Representations of the Drazin inverse on solution of a class singular differential equations, Linear and multilinear algebra, 59, 863-877, (2011) · Zbl 1227.15002  Campbell, S.L., The Drazin inverse and systems of second order linear differential equations, Linear and multilinear algebra, 14, 195-198, (1983) · Zbl 0523.15007  Campbell, S.L.; Meyer, C.D., Generalized inverse of linear transformations, (1979), Pitman London, (SIAM, Philadelphia, 2009)  Catral, M.; Olesky, D.D.; van den Driessche, P., Block representations of the Drazin inverse of a bipartite matrix, Electron. J. linear algebra, 18, 98-107, (2009) · Zbl 1183.15005  Catral, M.; Olesky, D.D.; van den Driessche, P., Graphical description of group inverses of certain bipartite matrices, Linear algebra appl., 432, 36-52, (2010) · Zbl 1184.15004  Deng, C.; Wei, Y., A note on the Drazin inverse of an anti-triangular matrix, Linear algebra appl., 431, 1910-1922, (2009) · Zbl 1177.15003  Deng, C.; Wei, Y., Representations for the Drazin inverse of $$2 \times 2$$ block matrix with singular Schur complement, Linear algebra appl., 435, 2766-2783, (2011) · Zbl 1225.15006  Eierman, M.; Marek, I.; Niethammer, W., On the solution of singular linear systems of algebraic equations by semi-iterative methods, Numer. math., 53, 265-283, (1988)  Hartwig, R.; Li, X.; Wei, Y., Representations for the Drazin inverse of $$2 \times 2$$ block matrix, SIAM J. matrix anal. appl., 27, 757-771, (2006) · Zbl 1100.15003  Lin, L.; Wei, Y.; Zhang, N., Convergence and quotient convergence of iterative methods for solving singular linear equations with index one, Linear algebra appl., 430, 1665-1674, (2009) · Zbl 1161.65026  Shader, B.L., Least square sign-solvability, SIAM J. matrix anal. appl., 16, 1056-1073, (1995) · Zbl 0837.05032  Shao, J.Y.; He, J.L., Matrices with doubly signed generalized inverses, Linear algebra appl., 355, 71-84, (2002) · Zbl 1021.15004  Shao, J.Y.; He, J.L.; Shan, H.Y., Matrices with special sign patterns of signed generalized inverses, SIAM J. matrix anal. appl., 24, 990-1002, (2003) · Zbl 1040.15006  Shao, J.Y.; Shan, H.Y., Matrices with signed generalized inverses, Linear algebra appl., 322, 105-127, (2001) · Zbl 0967.15002  Sidi, A., A unified approach to Krylov subspace methods for the Drazin-inverse solution of singular nonsymmetric linear systems, Linear algebra appl., 298, 99-113, (1999) · Zbl 0983.65054  Wei, Y., Expressions for the Drazin inverse of a $$2 \times 2$$ block matrix, Linear and multilinear algebra, 45, 131-146, (1998) · Zbl 0984.15004  Wei, Y., Index splitting for the Drazin inverse and the singular linear system, Appl. math. comput., 95, 115-124, (1998) · Zbl 0942.15003  Wei, Y., A characterization and representation of the generalized inverse $$A_{T, S}^{(2)}$$ and its applications, Linear algebra appl., 280, 87-96, (1998) · Zbl 0934.15003  Wei, Y.; Wu, H., Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index, J. comput. appl. math., 114, 305-318, (2000) · Zbl 0959.65046  Wei, Y.; Diao, H., Condition number for the Drazin inverse and the Drazin-inverse solution of singular linear system with their condition numbers, J. comput. appl. math., 182, 270-289, (2005) · Zbl 1077.15007  Wei, Y.; Li, X.; Bu, F.; Zhang, F., Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices-application of perturbation theory for simple invariant subspaces, Linear algebra appl., 419, 765-771, (2006) · Zbl 1151.15306  Wei, Y.; Deng, C., A note on additive results for the Drazin inverse, Linear and multilinear algebra, 59, 1319-1329, (2011) · Zbl 1237.15009  Xu, Q.; Song, C.; Wei, Y., The stable perturbation of the Drazin inverse of the square matrices, SIAM J. matrix anal. appl., 31, 1507-1520, (2010) · Zbl 1209.15009  Xu, Q.; Wei, Y.; Song, C., Explicit characterization of the Drazin index, Linear algebra appl., 436, 2273-2298, (2012) · Zbl 1236.15013  Zhou, J.; Bu, C.; Wei, Y., Group inverse for block matrices and some related sign analysis, Linear and multilinear algebra, 60, 669-681, (2012) · Zbl 1246.15009
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.