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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### King : A note on Drazin inverses.

Holomorphic and meromorphic properties of the g -Drazin inverse. Now, we begin with the following theorem. Sign up or log in Sign up using Google.

Let with and be nilpotent with. Ifthen and. 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.

So, by Lemma 2.

Lemma 1 see [ 10Theorem 2. If there exist two positive integers and such that andthen If is a perturbation ofthen, we have the following result in which has an upper bound.

Fanbin Bu and Yimin Wei, The algorithm for computing the Drazin inverses of two-variable polynomial matricesApplied mathematics and computation Finally, we give an example to illustrate our results.

So, by Lemmas 2. Assume that x and y are represented as. Before the theorem, let us recall that ifthen is invertible and. Applying Theorem 4we get. Assume that it holds forthat is.

### On the Drazin Inverse of the Sum of Two Matrices

Drazin inverse of a matrix Ask Question. This question on MathOverflow might be of interest. It is obvious thatand Sinceis invertible and By 3.

Since is nilpotent and is invertible, by Lemma 2. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

If a is quasinilpotent, we can apply Theorem 3 and we obtain 15 for this particular case. TroyHaskin Ok, I’ll check it out. The first one is this: The paper is organized as follows. 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.

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Some results on the generalized Drazin inverse of operator matrices. Then, i ii Proof.

Applying Theorem 4 to the particular case, we get. Ifthen is invertible and. Then, for any positive integerwhere the binomial coefficient.

In particular, ifis invertible and see, e.

## Journal of Applied Mathematics

We deduce the explicit expressions for and of two matrices and under the conditions and. So from andit follows, respectively, that and that Sinceand then. On the generalized Drazin inverse and generalized resolvent. Whenby iwe have Hence, by induction, we have for any. The representations of the Drazin inverse of differences of two matrices.