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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Assume that x and y are represented as. For interesting properties of the generalized Drazin inverse see [ 2 — 6 ]. Some Lemmas In this section, we will make preparations for discussing the Drazin inverse of the sum of two matrices in next section. Stack Drzin works best with JavaScript enabled.

Sinceby Lemma 2.

Letand withand define Then, Lemma 2. Journal List ScientificWorldJournal v.

## Journal of Applied Mathematics

On the other hand, it is easy to get that. The proof is very long and a cut and paste of its is not correct. If with andthen. Before the theorem, let us recall that ifthen is invertible and. Introduction The symbol stands for the set of complex matrices, and for short stands for the identity matrix.

Ifthen is invertible and. If there exists the generalized Drazin inverse, then the iinverse Drazin inverse of a is unique and is denoted by a d.

So, by Lemma 2. Applying Theorem 3we get. A generalized Drazin inverse. For a complete treatment of the generalized Drazin inverse, see [ 7Chapter 2].

### King : A note on Drazin inverses.

For example, in [ 7 ], the conditions are andin [ 9 ] they are andand in [ 15 ], they are and. Group inverses and Drazin inverses of ingerse and triangular Toeplitz matrices. Using the case of Theorem 3we get the following results.

In recent years, the Drazin inverse of the sum of two matrices or operators has been extensively investigated under different conditions see, [ 5 — 15 ]. To this end, we will introduce some lemmas. Then, for any positive integerwhere the binomial coefficient. The following theorem is our main result, and Theorem 3.

Lemma 1 see [ 10Theorem 2. Hence from 69 we obtain. Drazin [ 2 ] proved that in associative ring when are Drazin invertible and.

## Drazin inverse

So from andit follows, respectively, that and that Sinceand then. So, by Lemmas 2.

New additive results for the g -Drazin inverse. Using 40 and Theorem 14we have the following result.

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

Applying Theorem 4we get. Author information Article notes Copyright and License information Disclaimer. Obviously, whenit holds by statement i. Hence from 58 and 60 we obtain. Some results on the generalized Drazin inverse of invetse matrices. If with andthen Proof. Fanbin Bu and Yimin Wei, The algorithm for computing the Drazin inverses of two-variable polynomial matricesApplied mathematics and computation Additive results for the generalized Drazin inverse in a Banach algebra.

Main Results First we start the following result which is proved in [ 8 ] for matrices, extended in [ 9 ] for a bounded linear operator and niverse [ 10 ] for arbitrary elements in a Banach algebra.