The reverse order law for the Drazin inverses of multiple matrix products
type: Item
instance of: scholarly article
title: The reverse order law for the Drazin inverses of multiple matrix products
zbMATH Open document ID: 0998.15010
DOI: 10.1016/S0024-3795(01)00607-3
published in: Linear Algebra and its Applications
publication date: 2002-06-27T00:00:00Z
review text: It is well known that the Drazin inverse has been widely applied to the theory of finite Markov chains and singular differential and difference equations. In a classic paper \textit{T. N. E. Greville} [SIAM Rev. 8, 518-521 (1966; Zbl 0143.26303)] gave necessary and sufficient conditions of the reverse order law for the Moore-Penrose inverse \((AB)^{+}= B^{+}A^{+}\) to hold for two complex matrices \(A\) and \(B\). In general, the reverse order law does not hold for the Drazin inverse, that is \((AB)^{D} \neq B^{D}A^{D}\). Drazin proved that \((AB)^{D} = B^{D}A^{D}\) holds under the condition \(AB = BA\). In the paper under review the author gives necessary and sufficient conditions for the \(n\) term reverse order law \((A_1A_2\cdots A_n)^{D} = A_n^DA_{n-1}^D \cdots A_2^D A_1^D\) in terms of some rank equality.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1760988
zbMATH Keywords: Drazin inverse
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: reverse order law
zbMATH Keywords: matrix equation
author: Guo-rong Wang
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
cites work: Q4061081
cites work: Q3207128
cites work: Note on the Generalized Inverse of a Matrix Product
cites work: Block generalized inverses
cites work: Q4050723
cites work: Q3837185
description: scientific article; zbMATH DE number 1760988