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On block triangular matrices with signed Drazin inverse. (English) Zbl 1349.15084
For a real matrix $$A \in \mathbb{R}^{n \times n}$$, the sign pattern matrix relative to $$A$$ denoted by $$\mathrm{sgn}(A)$$ is the matrix whose entries are $$+$$, $$-$$ and $$0$$ obtained from $$A$$ by replacing each of its entry by its sign. Let $$\mathcal{Q}(A)$$ denote the set of all real matrices $$B$$ such that $$\mathrm{sgn}(B)=\mathrm{sgn}(A)$$. Let us recall that the index of a square matrix $$A$$ is the smallest nonnegative integer $$k$$ for which the ranks of $$A^k$$ and $$A^{k+1}$$ are the same. By a finite dimensionality argument, it follows that the index always exists. Now, let $$k$$ be the index of $$A$$. It is well known that there is a unique matrix $$X$$ such that $$A^{k+1}X=A^k$$, $$XAX=X$$ and $$AX=XA$$. Such an $$X$$ is unique, is called the Drazin inverse of $$A$$ and is denoted by $$A^D$$. We say that $$A$$ has a signed Drazin inverse if $$\mathrm{sgn}(C^D)=\mathrm{sgn}(A^D)$$ for any $$C \in \mathcal{Q}(A)$$. Finally, given $$A$$, we say that $$\mathrm{sgn}(A)$$ is potentially nilpotent if there exists a nilpotent matrix $$C \in \mathcal{Q}(A)$$. In the work under review, the authors present characterizations for certain block triangular matrices to have signed Drazin inverses. Let us only provide a sample result: Let $$M$$ be an upper triangular block square matrix which has a signed Drazin inverse and whose diagonal blocks are $$A$$ and $$C$$, where $$C$$ is potentially nilpotent. Then, $$M$$ is permutationally similar to a block square matrix whose upper triangular part is given by a matrix $$F$$ and whose bottom right block is permutationally similar to $$C$$ and is strictly upper triangular. Among other things, the authors show that $$\mathrm{sgn}((C^D)^2x^1) = \mathrm{sgn} ((A^D)^2f^1)$$, for all $$C \in \mathcal{Q}(A)$$ and $$X \in \mathcal{Q}(F)$$, where $$x^1$$ and $$f^1$$ denote the first columns of $$X$$ and $$F$$, respectively.

MSC:
 15B35 Sign pattern matrices 15A09 Theory of matrix inversion and generalized inverses
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References:
 [1] R. A. Brualdi, K. L. Chavey, B. L. Shader: Bipartite graphs and inverse sign patterns of strong sign-nonsingular matrices. J. Comb. Theory, Ser. B 62 (1994), 133–150. · Zbl 0807.05053 [2] R. A. Brualdi, H. J. Ryser: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39, Cambridge University Press, Cambridge, 1991. · Zbl 0746.05002 [3] R. A. Brualdi, B. L. Shader: Matrices of Sign-Solvable Linear Systems. Cambridge Tracts in Mathematics 116, Cambridge University Press, Cambridge, 1995. · 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. [5] M. Catral, D. D. Olesky, P. van den Driessche: Graphical description of group inverses of certain bipartite matrices. Linear Algebra Appl. 432 (2010), 36–52. · Zbl 1184.15004 [6] C. A. Eschenbach, Z. Li: Potentially nilpotent sign pattern matrices. Linear Algebra Appl. 299 (1999), 81–99. · Zbl 0941.15012 [7] B. L. Shader: Least squares sign-solvability. SIAM J. Matrix Anal. Appl. 16 (1995), 1056–1073. · Zbl 0837.05032 [8] J. -Y. Shao, J. -L. He, H. -Y. Shan: Matrices with special patterns of signed generalized inverses. SIAM J. Matrix Anal. Appl. 24 (2003), 990–1002. · Zbl 1040.15006 [9] J. -Y. Shao, Z. -X. Hu: Characterizations of some classes of strong sign nonsingular digraphs. Discrete Appl. Math. 105 (2000), 159–172. · Zbl 0965.05050 [10] J. -Y. Shao, H. -Y. Shan: Matrices with signed generalized inverses. Linear Algebra Appl. 322 (2001), 105–127. · Zbl 0967.15002 [11] C. Thomassen: When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse. Linear Algebra Appl. 119 (1989), 27–34. · Zbl 0673.05067 [12] J. Zhou, C. Bu, Y. Wei: Group inverse for block matrices and some related sign analysis. Linear and Multilinear Algebra 60 (2012), 669–681. · Zbl 1246.15009 [13] J. Zhou, C. Bu, Y. Wei: Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437 (2012), 1779–1792. · Zbl 1259.15008
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