## Journal of Applied Mathematics

### The Explicit Expression of the Drazin Inverse and Its Application

#### Abstract

We give explicit expressions of ${\left(P±Q\right)}_{d}$ of two matrices $P$ and $Q$, in terms of $P$, $Q$, ${P}_{d}$, and ${Q}_{d}$, ${\left(P±I\right)}_{d}$, under the condition that $PQ=P$, and apply the result to finding an explicit representation for the Drazin inverse of some $2×2$ block matrix.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 479260, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808126

Digital Object Identifier
doi:10.1155/2013/479260

Mathematical Reviews number (MathSciNet)
MR3100835

Zentralblatt MATH identifier
06950697

#### Citation

Liu, Xiaoji; Huang, Shaowu; Xu, Liang; Yu, Yaoming. The Explicit Expression of the Drazin Inverse and Its Application. J. Appl. Math. 2013 (2013), Article ID 479260, 7 pages. doi:10.1155/2013/479260. https://projecteuclid.org/euclid.jam/1394808126

#### References

• N. Castro-González, E. Dopazo, and M. F. Martínez-Serrano, “On the Drazin inverse of the sum of two operators and its application to operator matrices,” Journal of Mathematical Analysisand Applications, vol. 350, no. 1, pp. 207–215, 2009.
• D. S. Cvetković-Ilić, D. S. Djordjević, and Y. Wei, “Additive results for the generalized Drazin inverse in a Banach algebra,” Linear Algebra and Its Applications, vol. 418, no. 1, pp. 53–61, 2006.
• R. E. Hartwig, G. Wang, and Y. Wei, “Some additive results on Drazin inverse,” Linear Algebra and Its Applications, vol. 322, no. 1–3, pp. 207–217, 2001.
• X. Liu, L. Xu, and Y. Yu, “The representations of the Drazin inverse of differences of two matrices,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3652–3661, 2010.
• A. S. Cvetković and G. V. Milovanović, “On Drazin inverse ofoperator matrices,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 331–335, 2011.
• G. Wang, Y. Wei, and S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing, China, 2004.
• C. Y. Deng, “The Drazin inverse of bounded operators with commutativity up to a factor,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 695–703, 2008.
• C. Y. Deng, “The Drazin inverses of sum and difference of idempotents,” Linear Algebra and Its Applications, vol. 430, no. 4, pp.1282–1291, 2009.
• S. L. Campbell and C. D. Meyer Jr., Generalized Inverse of Linear Transformations, Pitman, Boston, Mass, USA, 1991.
• M. P. Drazin, “Pseudo-inverses in associative rings and semigroups,” The American Mathematical Monthly, vol. 65, pp. 506–514, 1958.
• C. D. Meyer Jr. and N. J. Rose, “The index and the Drazin inverseof block triangular matrices,” SIAM Journal on Applied Mathematics, vol. 33, no. 1, pp. 1–7, 1977.