DRAZIN INVERSE PDF

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 Moreover, if is nilpotent of indexthen is invertible and.

Hence from 69 we obtain. WhenHence 2. Giacomo Alessandroni 1 12 Before the theorem, let us recall that ifthen is invertible and Theorem 3.

Moreover, ifthen. Letand withand define Then. Some Lemmas In this section, we will make preparations for discussing the Drazin inverse of the sum of two matrices in next section.

Using the case of Theorem 3we get the following results.

From 44 and 47 it follows Published online Feb 1. The symbol stands for the set of complex matrices, and for short stands for the identity matrix. Author information Article notes Copyright and License information Disclaimer.

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Thus, is invertible and so isand Lemma 2. Therefore, assume thatand, without loss of generality, letinvers is invertible and is nilpotent of index. Subscribe to Table of Contents Alerts. Introduction Let A be a complex Banach algebra with unite 1.

Lemma 2 see [ 11Lemma 2. In Section 2we will deduce some lemmas. From drazn38and 39 it follows Whenby iwe have Hence, by induction, we have for any. It is well known that if is nilpotent, then. If are nilpotent with andthen taking in 2.

matlab – Drazin inverse of a matrix – Stack Overflow

Lemma 6 see [ 17Lemma 3. Linear Algebra and Its Applications. Acknowledgments The authors would like to thank the referees for their helpful comments and suggestions.

Applied Mathematics and Computation. Applying Theorem 4we get. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

In this section, we will give the explicit expressions for andunder the conditions and. 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.

Additive perturbation results for the Drazin inverse. Hence, by Onverse 2.

For interesting properties of the generalized Drazin inverse see [ 2 — 6 ]. Sign up or log in Sign up using Google. Received Sep 26; Accepted Jan 8. Drazin inverse of a matrix Ask Question. Assume that it holds forthat is. 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 dtazin website is subject to these policies.

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Journal of Applied Mathematics

Journal of Mathematical Analysis and Applications. So from andit follows, respectively, that and that Sinceand then.

From 7780and 81 it follows Then, Thus By the above argument, we can get 3. Observe that 50 and 51 yield.

Post Your Answer Discard 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.

Fanbin Bu and Yimin Wei, The algorithm for computing the Drazin inverses of two-variable polynomial matricesApplied mathematics and computation Journal of Applied Mathematics. Then, and Moreover, if is nilpotent of indexthen. Then substituting 20 and 24 in 22we get.