For a matrix A ∈ Cn×n with index 1, the group inverse A. # is the unique solution of the and its unique solution is the Drazin inverse, or. {1k,2,5}–inverse, of A. PDF | The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated. 1. Introduction. The main theme of this paper can be described as a study of the Drazin inverse. In , Drazin [4] introduced a different kind of generalized.

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The first one is this: Additive results for the generalized Drazin inverse in a Banach algebra. Proof — If a is quasinilpotent, we can apply Theorem 3 and we obtain 15 for this particular case. Then substituting 20 and 24 in 22we get. In Section 3we will present the explicit expressions for and of two matrices and under the conditions and. Before the theorem, let us recall that ifthen is invertible and.

Ifthen Moreover, if is nilpotent of indexthen is invertible and. Lemma 1 see [ 10Theorem 2. The index and the Drazin inverse of block triangular matrices. The set of all generalized Drazin invertible elements of A is denoted by A d.

Sign up using Facebook. So, by Lemmas 2.

Let withwhere is invertible and is nilpotent of indexand let be partitioned conformably with. The following theorem is our main result, and Theorem 3. Main Results First we start the following result which is proved in [ 8 ] for matrices, extended in [ 9 ] for a bounded linear operator and in [ 10 ] for arbitrary elements in a Banach algebra.

Journal of Mathematical Analysis and Applications. For example, in [ 7 ], the conditions are andin [ 9 ] they are andand in [ 15 ], they are and.

Ifthen, for any positive integersiii. We also writepartitioned conformably with. From 62 and 64 it follows Moreover, ifthen Proof. These results motivate us to investigate how to explicitly express the Drazin inverse of the sum under the conditions andwhich are implied by the condition.

Hence from 69 we obtain. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ifthen is invertible, and therefore. Thus, is invertible and so isand.

### matlab – Drazin inverse of a matrix – Stack Overflow

Now, we begin with the following theorem. Letand withand define Then, Lemma 2.

ingerse Thus, we have the following corollary. The following result is a generalization of [ 10Corollary 3. Introduction Let A be a complex Banach algebra with unite 1.

## Drazin inverse

So from andit follows, respectively, that and that Sinceand then. Similar to the discussion of 3. Since andwe can writepartitioned conformably withby Lemma 2. Since is nilpotent and is invertible, by Lemma 2.

Some additive results on Drazin inverse. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Recall that for withthere exists an nonsingular matrix such that where is a nonsingular matrix and is nilpotent of indexand see [ 13 ]. While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Additive results for the generalized Drazin inverse. Lemma 6 see [ 17Lemma 3.