Abstract
For complex matrices, we know from Chap. 3 that the group inverse only exists for matrices with index no more than 1, so it is not suitable for the more common class of matrices whose index exceed 1. In this chapter, we introduce the Drazin inverse which is a slight generalization of the group inverse.
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References
N. Castro-González, J.J. Koliha, Y.M. Wei, Perturbation of the Drazin inverse for matrices with equal eigenprojections at zero. Linear Algebra Appl. 312, 181–189 (2000)
N. Castro-González, J.J. Koliha, I. Straškraba, Perturbation on the Drazin inverse. Soochow J. Math. 27, 201–211 (2001)
N. Castro-González, C. Mendes-Araújo, P. Patrício, Generalized inverses of a sum in rings. Bull. Austral. Math. Soc. 82, 156–164 (2010)
J.L. Chen, A note on generalized inverses of a product. Northeast Math. J. 12, 431–440 (1996)
J.L. Chen, H.H. Zhu, Drazin invertibility of product and difference of idempotents in a ring. Filomat 28(6), 1133–1137 (2014)
J.L. Chen, G.F. Zhuang, Y.M. Wei, The Drazin inverse of a sum of morphisms. Acta Math. Sci. Ser. A 29(3), 538–552 (2009)
R.E. Cline, An application of representation for the generalized inverse of a matrix. MRC Technical Report 592, 1965
D.S. Cvetković-Ilić, Some results on the (2,2,0) Drazin inverse problem. Linear Algebra Appl. 438, 4726–4741 (2013)
D.S. Cvetković-Ilić, C.Y. Deng, Some results on the Drazin invertibility and idempotents. J. Math. Anal. Appl. 359, 731–738 (2009)
M.P. Drazin, Pseudo-inverses in associative rings and semigroups. Amer. Math. Mon. 65, 506–514 (1958)
J.J. Koliha, P. Patrício, Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72, 137–152 (2002)
J.J. Koliha, D.S. Cvetković-Ilić, C.Y. Deng, Generalized Drazin invertibility of combinations of idempotents. Linear Algebra Appl. 437, 2317–2324 (2012)
P. Patrício, R. Puystjens, Drazin-Moore-Penrose invertibility in rings. Linear ALgebra Appl. 389, 159–173 (2004)
P. Patrício, A. Veloso Da Costa, On the Drazin index of regular elements. Cent. Eur. J. Math. 7(2), 200–205 (2009)
R. Puystjens, M.C. Gouveia, Drazin invertibility for matrices over an arbitrary ring. Linear Algebra Appl. 385, 105–116 (2004)
Y.M. Wei, Perturbations bound of the Drazin inverse. Appl Math Comput. 125, 231–244 (2002)
Y.M. Wei, G.R. Wang, The perturbation theory for the Drazin inverse and its applications. Linear Algebra Appl. 258, 179–186 (1997)
H. You, J.L. Chen, Generalized inverses of a sum of morphisms. Linear Algebra Appl. 338, 261–273 (2001)
H. You, J.L. Chen, The Drazin inverse of a morphism in additive category. (Chinese) J. Math. (Wuhan) 22(3), 359–364 (2002)
G.F. Zhuang, J.L. Chen, D.S. Cvetković-Ilić, Y.M. Wei, Additive property of Drazin invertibility of elements in a ring. Linear Multilinear Algebra 60(8), 903–910 (2012)
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Chen, J., Zhang, X. (2024). Drazin Inverses. In: Algebraic Theory of Generalized Inverses. Springer, Singapore. https://doi.org/10.1007/978-981-99-8285-1_4
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DOI: https://doi.org/10.1007/978-981-99-8285-1_4
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