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Further Expressions on the Drazin Inverse for Block Matrix

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Abstract

This article addresses the problem of developing new expressions for the Drazin inverse of complex block matrix \(M=\left( \begin{array}{cc} A &{} B \\ C &{} D \\ \end{array} \right) \in {\mathbb {C}}^{n\times n}\) (where A and D are square matrices but not necessarily of the same size) in terms of the Drazin inverse of matrix A and of its generalized Schur complement \(S=D-CA^DB\) which is not necessarily invertible. This formula is the extension of the well-known Banachiewicz inversion formula of complex block matrix M. In addition, we provide representation for the Drazin inverse of complex block matrix M without any restriction on the generalized Schur complement S and under different conditions than those used in some current literature on this subject. Finally, several illustrative numerical examples are considered to demonstrate our results.

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References

  • Banachiewicz T (1937) Zur Berechnung der Determinaten wie auch der Inversen und zur darauf basierten Auflösung der Systeme linearer Gleichungen. Acta Astron Sér C 3:41–67

    MATH  Google Scholar 

  • Ben-Israel A, Greville TNE (2003) Generalized inverses: theory and applications, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Campbell SL (1980) Singular systems of differential equations. Pitman, London

    MATH  Google Scholar 

  • Campbell SL (1983) The Drazin inverse and systems of second order linear differential equations. Linear Multilinear Algebra 14:195–198

    Article  Google Scholar 

  • Campbell SL, Meyer CD (1991) Generalized inverse of linear transformations. Dover, New York

    MATH  Google Scholar 

  • Hartwig RE, Wang G, Wei Y (2001) Some additive results on Drazin inverse. Linear Algebra Appl 322:207–217

    Article  MathSciNet  Google Scholar 

  • Hartwig RE, Li X, Wei Y (2006) Representations for the Drazin inverse of \(2\times 2\) block matrix. SIAM J Matrix Anal Appl 27:757–771

    Article  Google Scholar 

  • Li X, Wei Y (2007) A note on the representations for the Drazin inverse of \(2\times 2\) block matrices. Linear Algebra Appl 423:332–338

    Article  MathSciNet  Google Scholar 

  • Ljubisavljević J, Cvetković-Ilić DS (2011) Additive results for the Drazin inverse of block matrices and applications. J Comput Appl Math 235:3683–3690

    Article  MathSciNet  Google Scholar 

  • Meyer CD, Rose NJ (1977) The index and the Drazin inverse of block triangular matrices. SIAM J Appl Math 33:1–7

    Article  MathSciNet  Google Scholar 

  • Miao J (1989) Results of the Drazin inverse of block matrices (in Chinese). J Shanghai Normal Univ 18:25–31

    Google Scholar 

  • Wang G, Wei Y, Qiao S (2004) Generalized inverses: theory and computations. Science Press, Beijing/New York

    MATH  Google Scholar 

  • Wei Y (1998) Expressions for the Drazin inverse of a \(2\times 2\) block matrix. Linear Multilinear Algebra 45:131–146

    Article  MathSciNet  Google Scholar 

  • Yang H, Liu X (2011) The Drazin inverse of the sum of two matrices and its applications. J Comput Appl Math 235:1412–1417

    Article  MathSciNet  Google Scholar 

  • Zhang F (ed) (2005) The Schur complement and its applications. Springer, Berlin

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, 64200, Pakistan

    Abdul Shakoor

  2. Basic Sciences and Humanities Department, University of Engineering and Technology Lahore, Faisalabad Campus, Faisalabad, 38000, Pakistan

    Ilyas Ali & Abdur Rehman

  3. School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Samad Wali

Authors
  1. Abdul Shakoor
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  2. Ilyas Ali
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  3. Samad Wali
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  4. Abdur Rehman
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Correspondence to Abdul Shakoor.

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Shakoor, A., Ali, I., Wali, S. et al. Further Expressions on the Drazin Inverse for Block Matrix. Iran J Sci Technol Trans Sci 44, 833–837 (2020). https://doi.org/10.1007/s40995-020-00883-7

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  • DOI: https://doi.org/10.1007/s40995-020-00883-7

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