Geodesic
On block triangular matrices with signed Drazin inverse
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 883-892 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-014-0141-6
Classification : 15A09, 15B35
Keywords: sign pattern matrix; signed Drazin inverse; strong sign nonsingular matrix 
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Bu, Changjiang; Wang, Wenzhe; Zhou, Jiang; Sun, Lizhu. On block triangular matrices with signed Drazin inverse. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 883-892. doi: 10.1007/s10587-014-0141-6

[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 | MR | Zbl

[2] Brualdi, R. A., Ryser, H. J.: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39 Cambridge University Press, Cambridge (1991). | MR | Zbl

[3] Brualdi, R. A., Shader, B. L.: Matrices of Sign-Solvable Linear Systems. Cambridge Tracts in Mathematics 116 Cambridge University Press, Cambridge (1995). | MR | Zbl

[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 | Zbl

[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

[6] Eschenbach, C. A., Li, Z.: Potentially nilpotent sign pattern matrices. Linear Algebra Appl. 299 (1999), 81-99. | MR | Zbl

[7] Shader, B. L.: Least squares sign-solvability. SIAM J. Matrix Anal. Appl. 16 (1995), 1056-1073. | DOI | MR | Zbl

[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 | MR

[9] Shao, J.-Y., Hu, Z.-X.: Characterizations of some classes of strong sign nonsingular digraphs. Discrete Appl. Math. 105 (2000), 159-172. | DOI | MR | Zbl

[10] Shao, J.-Y., Shan, H.-Y.: Matrices with signed generalized inverses. Linear Algebra Appl. 322 (2001), 105-127. | DOI | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[13] Zhou, J., Bu, C., Wei, Y.: Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437 (2012), 1779-1792. | MR | Zbl

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