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A note on the representation for the Drazin inverse of $2 \times 2$ block matrices. (English) Zbl 1148.15001
The author gives the representations of the Drazin inverse of $2 \times 2$ block matrices under conditions weaker than those used in the following papers on the subject: 1. {\it D. S. Djordjević} and {\it P. S. Stanimirović} [Czech. Math. J. 51, No. 3, 617--634 (2001; Zbl 1079.47501)]; 2. {\it R. Hartwig, X. Li} and {\it Y. Wei} [SIAM J. Matrix Anal. Appl. 27, No. 3, 757--771 (2006; Zbl 1100.15003)].

15A09Matrix inversion, generalized inverses
Full Text: DOI
[1] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses: theory and applications, (2003) · Zbl 1026.15004
[2] Campbell, S. L.: Singular systems of differential equations, (1980) · Zbl 0444.34062 · doi:10.1080/00036818008839326
[3] Campbell, S. L.: The Drazin inverse and systems of second order linear differential equations, Linear and multilinear algebra 14, 195-198 (1983) · Zbl 0523.15007 · doi:10.1080/03081088308817556
[4] Campbell, S. L.; Meyer, C. D.: Generalized inverse of linear transformations, (1979) · Zbl 0417.15002
[5] Castro-González, N.; Dopazo, E.; Robles, J.: Formulas for the Drazin inverse of special block matrices, Appl. math. Comput. 174, 252-270 (2006) · Zbl 1097.15005 · doi:10.1016/j.amc.2005.03.027
[6] Castro-González, N.: Additive perturbation results for the Drazin inverse, Linear algebra appl. 397, 279-297 (2005) · Zbl 1071.15003 · doi:10.1016/j.laa.2004.11.001
[7] González, N. Castro; Koliha, J. J.: New additive results for the g-Drazin inverse, Proc. roy. Soc. Edinburgh sect. A 134, 1085-1097 (2004) · Zbl 1088.15006 · doi:10.1017/S0308210500003632
[8] Chen, X.; Hartwig, R. E.: The group inverse of a triangular matrix, Linear algebra appl. 237 -- 238, 97-108 (1996) · Zbl 0851.15005 · doi:10.1016/0024-3795(95)00561-7
[9] Cvetković-Ilić, D. S.; Djordjević, D. S.; Wei, Y.: Additive results for the generalized Drazin inverse in a Banach algebra, Linear algebra appl. 418, 53-61 (2006) · Zbl 1104.47040 · doi:10.1016/j.laa.2006.01.015
[10] Djordjević, D. S.; Stanimirović, P. S.: On the generalized Drazin inverse and generalized resolvent, Czechoslovak math. J. 51, No. 126, 617-634 (2001) · Zbl 1079.47501 · doi:10.1023/A:1013792207970
[11] Drazin, M. P.: Pseudoinverse in associative rings and semigroups, Amer. math. Monthly 65, 506-514 (1958) · Zbl 0083.02901 · doi:10.2307/2308576
[12] Hartwig, R. E.; Shoaf, J. M.: Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices, Austral. J. Math. 24, No. A, 10-34 (1977) · Zbl 0372.15003
[13] Hartwig, R.; Li, X.; Wei, Y.: Representations for the Drazin inverse of $2\times 2$ block matrix, SIAM J. Matrix anal. Appl. 27, 757-771 (2006) · Zbl 1100.15003 · doi:10.1137/040606685
[14] Hartwig, R. E.; Wang, G.; Wei, Y.: Some additive results on Drazin inverse, Linear algebra appl. 322, 207-217 (2001) · Zbl 0967.15003 · doi:10.1016/S0024-3795(00)00257-3
[15] Meyer, C. D.; Rose, N. J.: The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. math. 33, 1-7 (1977) · Zbl 0355.15009 · doi:10.1137/0133001
[16] Wang, G.; Wei, Y.; Qiao, S.: Generalized inverses: theory and computations, (2003)
[17] Li, X.; Wei, Y.: A note on the representations for the Drazin inverse of $2\times 2$ block matrices, Linear algebra appl. (2007) · Zbl 1121.15008 · doi:10.1016/j.laa.2007.01.005
[18] Wei, Y.: Expression for the Drazin inverse of a $2\times 2$ block matrix, Linear and multilinear algebra 45, 131-146 (1998)
[19] Wei, Y.; Li, X.; Bu, F.; Zhang, F.: Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices -- application of perturbation theory for simple invariant subspaces, Linear algebra appl. 419, 765-771 (2006) · Zbl 1151.15306 · doi:10.1016/j.laa.2006.06.015