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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
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##### References:
 [1] Brualdi, R.A.; Shader, B.L., Matrices of sign-solvable linear systems, (1995), Cambridge University Press Cambridge · Zbl 0833.15002 [2] 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 [3] 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 [4] Campbell, S.L.; Meyer, C.D., Generalized inverse of linear transformations, (1979), Pitman London, (SIAM, Philadelphia, 2009) [5] 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 [6] 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 [7] 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 [8] 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 [9] 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) [10] 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 [11] 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 [12] Shader, B.L., Least square sign-solvability, SIAM J. matrix anal. appl., 16, 1056-1073, (1995) · Zbl 0837.05032 [13] Shao, J.Y.; He, J.L., Matrices with doubly signed generalized inverses, Linear algebra appl., 355, 71-84, (2002) · Zbl 1021.15004 [14] 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 [15] Shao, J.Y.; Shan, H.Y., Matrices with signed generalized inverses, Linear algebra appl., 322, 105-127, (2001) · Zbl 0967.15002 [16] 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 [17] 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 [18] Wei, Y., Index splitting for the Drazin inverse and the singular linear system, Appl. math. comput., 95, 115-124, (1998) · Zbl 0942.15003 [19] 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 [20] 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 [21] 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 [22] 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 [23] Wei, Y.; Deng, C., A note on additive results for the Drazin inverse, Linear and multilinear algebra, 59, 1319-1329, (2011) · Zbl 1237.15009 [24] 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 [25] Xu, Q.; Wei, Y.; Song, C., Explicit characterization of the Drazin index, Linear algebra appl., 436, 2273-2298, (2012) · Zbl 1236.15013 [26] 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
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