# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  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  R. A. Brualdi, H. J. Ryser: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39, Cambridge University Press, Cambridge, 1991. · Zbl 0746.05002  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  S. L. Campbell, C. D. Meyer, Jr.: Generalized Inverses of Linear Transformations. Surveys and Reference Works in Mathematics 4, Pitman Publishing, London, 1979.  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  C. A. Eschenbach, Z. Li: Potentially nilpotent sign pattern matrices. Linear Algebra Appl. 299 (1999), 81–99. · Zbl 0941.15012  B. L. Shader: Least squares sign-solvability. SIAM J. Matrix Anal. Appl. 16 (1995), 1056–1073. · Zbl 0837.05032  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  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  J. -Y. Shao, H. -Y. Shan: Matrices with signed generalized inverses. Linear Algebra Appl. 322 (2001), 105–127. · Zbl 0967.15002  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  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  J. Zhou, C. Bu, Y. Wei: Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437 (2012), 1779–1792. · Zbl 1259.15008
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.