Full-rank and determinantal representation of the Drazin inverse
type: Item
instance of: scholarly article
title: Full-rank and determinantal representation of the Drazin inverse
zbMATH Open document ID: 0956.15005
DOI: 10.1016/S0024-3795(00)00075-6
author: Predrag S. Stanimirović
author: Dragan S. Djordjević
published in: Linear Algebra and its Applications
publication date: 2000-08-29T00:00:00Z
review text: 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.
reviewed by: Lubomír Bakule
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1462180
zbMATH Keywords: Drazin inverse
zbMATH Keywords: full-rank factorization
zbMATH Keywords: determinantal representation
zbMATH Keywords: proportional minors
zbMATH Keywords: full-rank representation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1462180