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A revisit to a reverse-order law for generalized inverses of a matrix product and its variations

Yongge Tian (yongge.tian***at***gmail.com)

Abstract: This paper approaches a fundamental reverse-order law $(AB)^{(1)} = B^{\dag}A^{\dag}$ in the theory of generalized inverse, where $A$ and $B$ are $m \times n$ and $n \times p$ complex matrices, respectively, $(AB)^{(1)}$ is $\{1\}$-inverse of $AB$, $A^{\dag}$ and $B^{\dag}$ are the Moore--Penrose inverses of $A$ and $B$, respectively. We will collect and derive many known and novel equivalent statements for the reverse-order law and its variations to hold by using the methodology of ranks and ranges of matrices.

Keywords: matrix product, Moore--Penrose inverse, reverse-order law, rank, range

Category 1: Global Optimization (Theory )

Citation:

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Entry Submitted: 08/24/2017
Entry Accepted: 08/24/2017
Entry Last Modified: 09/11/2017

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