## Full-rank and determinantal representation of the Drazin inverse.(English)Zbl 0956.15005

Consider a complex $$m\times n$$ matrix $$A$$ satisfying the equations in $$G: GAG=G$$, $$AG=GA$$ and $$A^{k+1}G=A^k$$ for a positive integer $$k= \text{ind}(A)= \min\{p: \text{rank} (A^{p+1})= \text{rank}(A^p)\}$$. Such a matrix $$G=A^D$$ is said to be the Drazin inverse (DI) of $$A$$.
A full-rank representation of the DI of a square $$A$$ is introduced by means of the components from an arbitrary full-rank factorization of any matrix power $$A^l$$, $$l\geq k= \text{ind}(A)$$. A determinantal representation of the DI which is derived using the general representation describes the elements of the DI as a fraction of two expressions involving the minors of the order $$\text{rank}(A^k)$$, $$k= \text{ind}(A)$$ taken from the matrix $$A$$ and the rank invariant powers $$A^l$$, $$l\geq k$$.
Necessary and sufficient conditions for the existence of the DI are proved for matrices whose elements are taken from an integral domain. Correlations between the minors of the order $$\text{rank}(A^k)$$ selected from matrices $$A^D$$, $$(A^D)^p$$, $$p\geq 1$$ and from the matrix $$A^k$$, $$k= \text{ind}(A)$$ are explicitely derived.

### MSC:

 15A09 Theory of matrix inversion and generalized inverses
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### References:

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