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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Ifthen Proof.

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### King : A note on Drazin inverses.

Now assume invsrse and, without loss of generality, can be written aswhere is invertible and is nilpotent of index. On the other hand, it is easy to get that. By induction on for 2. So, by Lemmas 2. Some Lemmas In this section, we will make preparations for discussing the Drazin inverse of the sum of two matrices in next section. Proceedings of the Royal Society of Edinburgh: The set of all generalized Drazin invertible elements of A is denoted by A d. If there exists the generalized Drazin inverse, then the generalized Drazin inverse of a is unique and is denoted by a d.

Received Sep 26; Accepted Jan 8. Drazin inverse of a matrix Ask Question. Obviously, whenit holds by statement i. If andthen Proof.

From 44 and 47 it follows If a is quasinilpotent, we can apply Theorem 3 and we obtain 15 for this particular case. Ifthen is invertible and. Introduction Let A be a complex Banach algebra with unite 1. Sign up using Facebook. The symbol stands for the set of complex matrices, and for short stands for the identity drazib. Linear and Multilinear Algebra.

## Drazin inverse

Some results on the generalized Drazin inverse of operator matrices. Consider the matrices We observe that andbut. Recall that for withthere exists an nonsingular matrix such that where is a nonsingular matrix and is nilpotent of indexand see [ 13 ].

Acknowledgments The authors would like to thank the referees for their helpful comments and suggestions. Support Center Support Center. Email Required, but never shown. 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. Letand withand define Then, Lemma 2. New additive results for the g -Drazin inverse.

So, by Lemma 2. Moreover, ifthen.

Applying Theorem 4we get. Assume that, forthe equation holds, that is. Since is nilpotent and is invertible, by Lemma 2. Let and be invertible. Then, for any positive integerwhere the binomial coefficient. We will show by induction that 2. In this section, we will give the explicit expressions for andunder the conditions and.

The proof is very long and a cut and paste of its is not correct. The authors declare that there is no conflict of interests regarding the publication of this paper. As an application we give some new representations for the generalized Drazin inverse of an operator matrix. For interesting properties of the generalized Drazin inverse see [ 2 — 6 ].

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

Before the theorem, let us recall that ifthen is invertible and Theorem 3. Giacomo Alessandroni 1 12 From 7780and 81 it follows TroyHaskin Ok, I’ll check it out.

Thus, is invertible and so isand.