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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### Drazin inverse – Wikipedia

Lemma 2 see [ 11Lemma 2. Author information Article notes Copyright and License information Disclaimer.

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Moreover, if are nilpotent with andthen is nilpotent and its index is less than. As an application we inversse some new representations for the generalized Drazin inverse of an operator matrix.

### King : A note on Drazin inverses.

Drazin inverse of a matrix Ask Question. Recall that for withthere exists an nonsingular matrix such that where is a nonsingular matrix and is nilpotent of indexand see [ 13 ]. These results motivate us to investigate how to invegse express the Drazin inverse of the sum under the conditions andwhich are implied by the condition. 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.

Applying Theorem 4 to the particular case, we get. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, invrese reproduction in any medium, provided the original work is properly cited.

In this section, we will give the explicit expressions for andunder the conditions and. Sign up using Email and Password. If there exists the generalized Drazin inverse, then the generalized Drazin inverse of a is unique and is denoted by a d. Then, i ii Proof. For a complete treatment of the generalized Drazin inverse, see [ 7Chapter 2]. Assume that, forthe equation holds, that is.

Then, for any positive integerwhere the binomial coefficient. From this, we can easily verify Therefore, ifthen. Fanbin Bu and Yimin Wei, The algorithm for computing the Drazin inverses of two-variable polynomial matricesApplied mathematics and computation From 62 and 64 it follows Consider the matrices We observe that andbut. In this section, we will make preparations for discussing the Drazin inverse of the sum of two matrices in next section.

The first one is this: From 44 and 47 it follows Proof — If a is quasinilpotent, we can apply Theorem 3 and we obtain 15 for this particular case. 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. Observe that 50 and 51 yield. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Yes, I have seen it, but since it seems to be quite old, I assumed there would be a MATLAB or a Mathematica code by now, that computes quite efficiently the Drazin inverse of a singular matrix.

We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. The following result is a generalization of [ 10Corollary 3. Introduction The symbol stands for the set of complex matrices, and for short stands for the identity matrix. Main Results In this section, we will give the explicit expressions for andunder the conditions and. Is there an algorithm that computes the Drazin inverse of a singular matrix?

Also, we give the upper bound of. The following theorem is our main result, and Theorem 3. Whenby iwe have Hence, by induction, we have for any.

## Drazin inverse

We also give the upper bound of. Ifthen Moreover, if is nilpotent of indexthen is invese and Proof. Using 40 and Theorem 12we have the following result.

Therefore, assume thatand, without loss of generality, letwhere is invertible and ddazin nilpotent of index. In [ 9Theorem 5. From 71 and 72 it follows Since is nilpotent and is invertible, by Lemma 2. Thus, we have the following corollary. In [ 4 ], Hartwig et al.