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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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