## An alternative limit expression of Drazin inverse and its application.(English)Zbl 0805.65041

Let $$A$$ be an $$n\times n$$ matrix and $$\text{ind}(A)$$ the index of $$A$$, the smallest nonnegative integer for which $$\text{rank}(A^ k)=\text{rank}(A^{k+1})$$. The author presents a new limit expression for the Drazin inverse
(1) $$A^ D=\lim_{\lambda\to 0} (\lambda+A)^{-(\ell+1)} A^ \ell$$, where $$\ell\geq k=\text{ind}(A)$$. It is assumed that $$- \lambda\not\in\sigma(A)$$, the set of all eigenvalues of $$A$$. The limit expression (1) provides a new proof of a finite algorithm for the Drazin inverse given by T. N. E. Greville [Linear Algebra Appl. 6, 205-208 (1973; Zbl 0247.15004)].

### MSC:

 65F20 Numerical solutions to overdetermined systems, pseudoinverses 15A09 Theory of matrix inversion and generalized inverses

### Keywords:

Drazin inverse; finite algorithm

Zbl 0247.15004
Full Text:

### References:

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