Czechoslovak Mathematical Journal

, Volume 64, Issue 4, pp 883–892 | Cite as

On block triangular matrices with signed Drazin inverse

  • Changjiang Bu
  • Wenzhe Wang
  • Jiang Zhou
  • Lizhu Sun


The sign pattern of a real matrix A, denoted by sgnA, is the (+, −, 0)-matrix obtained from A by replacing each entry by its sign. Let Q(A) denote the set of all real matrices B such that sgnB = sgnA. 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 \(\operatorname{sgn} {\tilde A^d} = \operatorname{sgn} {A^d}\) for any \(\tilde A \in Q(A)\), where A d denotes the Drazin inverse of A. In this paper, we give necessary conditions for some block triangular matrices to have signed Drazin inverse.


sign pattern matrix signed Drazin inverse strong sign nonsingular matrix 

MSC 2010

15B35 15A09 


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  1. [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.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    R. A. Brualdi, H. J. Ryser: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39, Cambridge University Press, Cambridge, 1991.CrossRefMATHGoogle Scholar
  3. [3]
    R. A. Brualdi, B. L. Shader: Matrices of Sign-Solvable Linear Systems. Cambridge Tracts in Mathematics 116, Cambridge University Press, Cambridge, 1995.CrossRefMATHGoogle Scholar
  4. [4]
    S. L. Campbell, C. D. Meyer, Jr.: Generalized Inverses of Linear Transformations. Surveys and Reference Works in Mathematics 4, Pitman Publishing, London, 1979.MATHGoogle Scholar
  5. [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.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    C. A. Eschenbach, Z. Li: Potentially nilpotent sign pattern matrices. Linear Algebra Appl. 299 (1999), 81–99.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    B. L. Shader: Least squares sign-solvability. SIAM J. Matrix Anal. Appl. 16 (1995), 1056–1073.CrossRefMATHMathSciNetGoogle Scholar
  8. [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.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    J. -Y. Shao, Z. -X. Hu: Characterizations of some classes of strong sign nonsingular digraphs. Discrete Appl. Math. 105 (2000), 159–172.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    J. -Y. Shao, H. -Y. Shan: Matrices with signed generalized inverses. Linear Algebra Appl. 322 (2001), 105–127.CrossRefMATHMathSciNetGoogle Scholar
  11. [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.CrossRefMATHMathSciNetGoogle Scholar
  12. [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.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    J. Zhou, C. Bu, Y. Wei: Some block matrices with signed Drazin inverses. Linear Algebra Appl. 437 (2012), 1779–1792.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014

Authors and Affiliations

  • Changjiang Bu
    • 1
  • Wenzhe Wang
    • 1
  • Jiang Zhou
    • 2
  • Lizhu Sun
    • 3
  1. 1.College of ScienceHarbin Engineering UniversityHarbinHeilongjiang Province, P.R. China
  2. 2.College of Computer Science and Technology, and College of ScienceHarbin Engineering UniversityHarbinHeilongjiang Province, P.R. China
  3. 3.School of ScienceHarbin Institute of TechnologyHarbinHeilongjiang Province, P.R. China

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