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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Letand withand define Then, Lemma 2. Journal of the Australian Mathematical Society A. If andthen Proof. The following result is a generalization of [ 10Corollary 3.

A note on the Drazin inverse of an anti-triangular matrix. A generalized Drazin inverse.

Some results on the generalized Drazin inverse of operator matrices. Introduction Let A be a complex Banach algebra with unite 1. If there exists the generalized Drazin inverse, then the generalized Drazin inverse of a is unique and is denoted by a d. draziin

## Journal of Applied Mathematics

The paper is organized as follows. Also, we give the upper bound of. Abstract We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. National Center for Biotechnology InformationU.

Thus, is invertible and so isand.

Forits Drazin inverse, denoted byis defined as the unique matrix satisfying where is the index of. Main Results In this section, we will give the explicit expressions for andunder the conditions and. Consider the matrices We observe that andbut. Then, and Moreover, if is nilpotent of indexthen.

Journal of the Australian Mathematical Society. Published online Feb 1.

### King : A note on Drazin inverses.

Then, for any positive integerwhere the binomial coefficient. We also give the upper bound of. New additive results for the g -Drazin inverse. Is invese an algorithm that computes the Drazin inverse of a singular matrix?

Linear Algebra and Its Applications. Inveerse Theorem 4 to the particular case, we get. Finally, we give an example to illustrate our results. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

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

Now assume that and, without loss of generality, can be written aswhere is invertible and is nilpotent of index. From 71 and 72 it follows Ifthen, for any positive integersiii.

Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices. On the Drazin inverse of the sum of two operators ihverse its application to operator matrices. The proof is very long and a cut and paste of its is not correct. Additive results for the generalized Drazin inverse. Proceedings of the Royal Society of Edinburgh: From this, we can easily verify Therefore, ifthen.

Lectures on Generalized Inverses. We will show by induction that 2. It is obvious thatand Sinceis invertible and By 3. Hence, by Lemma 2.