Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-18T15:11:55.906Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED INVERSES OF A SUM IN RINGS

Part of: Basic linear algebra

Published online by Cambridge University Press:  22 April 2010

N. CASTRO-GONZÁLEZ ,
C. MENDES-ARAÚJO  and
PEDRO PATRICIO
N. CASTRO-GONZÁLEZ*
Affiliation:
Facultad de Informática, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain (email: [email protected])
C. MENDES-ARAÚJO
Affiliation:
Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal (email: [email protected])
PEDRO PATRICIO
Affiliation:
Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1−ba in terms of generalized inverses of 1−ab. In our development we prove that the Drazin index of 1−ba is equal to the Drazin index of 1−ab.

Keywords

regular elementreflexive inverseDrazin indexDrazin inverseEP elements

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 156 - 164
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author was partially supported by Project MTM2007-67232, ‘Ministerio de Educación y Ciencia’ of Spain. The second and third authors were supported by the Portuguese Foundation for Science and Technology-FCT through the POCTI research program.

References

[1]Barnes, B. A., ‘Common operator properties of the linear operators RS and SR’, Proc. Amer. Math. Soc. 126 (1998), 1055–1061.CrossRefGoogle Scholar
[2]Ben-Israel, A. and Greville, T. N. E., Generalized Inverses. Theory and Applications, 2nd edn (Springer, New York, 2003).Google Scholar
[3]Campbell, S. L. and Meyer, C. D. Jr, Generalized Inverses of Linear Transformations (Pitman, London, 1979); (Dover, New York, 1991).Google Scholar
[4]Castro-González, N. and Vélez-Cerrada, J. Y., ‘Elements of rings and Banach algebras with related spectral idempotents’, J. Aust. Math. Soc. 80 (2006), 383–396.CrossRefGoogle Scholar
[5]Hartwig, R. E., ‘Block generalized inverses’, Arch. Ration. Mech. Anal. 61 (1976), 197–251.CrossRefGoogle Scholar
[6]Hartwig, R. E. and Shoaf, J., ‘Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices’, J. Aust. Math. Soc. Ser. A 24 (1977), 10–34.CrossRefGoogle Scholar
[7]Koliha, J. J. and Patricio, P., ‘Elements of rings with equal spectral idempotents’, J. Aust. Math. Soc. 72 (2002), 137–152.CrossRefGoogle Scholar
[8]Patricio, P. and Veloso da Costa, A., ‘On the Drazin index of regular elements’, Cent. Eur. J. Math. 7(2) (2009), 200–205.Google Scholar
[9]Pearl, M. H., ‘Generalized inverses of matrices with entries taken from an arbitrary field’, Linear Algebra Appl. 1 (1968), 571–587.CrossRefGoogle Scholar
[10]Puystjens, R. and Hartwig, R. E., ‘The group inverse of a companion matrix’, Linear Multilinear Algebra 43 (1997), 137–150.CrossRefGoogle Scholar