International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 8, Pages 505-509
doi:10.1155/S0161171202007317
On the relation between Moore's and Penrose's conditions
Department of Mathematics and Statistics, University of Missouri-Rolla, 1870 Miner Circle, Rolla 65409-0020, MO, USA
Received 13 April 2001
Copyright © 2002 Gaoxiong Gan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Moore (1920) defined the reciprocal of any matrix over
the complex field by three conditions, but the beauty of the
definition was not realized until Penrose (1955) defined the same
inverse using four conditions. The reciprocal is now often called
the Moore-Penrose inverse, and has been widely used in
various areas. This note comments on the definitions of
Moore-Penrose inverse, and gives a new characterization for two
types of weak Moore-Penrose inverses, which exposes an
important relation between Moore's and Penrose's
conditions. It also attempts to emphasize the merit of Moore's
definition, which has been overlooked mainly due to Moore's
unique notation. Two examples are given to demonstrate some
combined applications of Moore's and Penrose's conditions,
including a correction for an incorrect proof of Ben-Israel's
(1986) characterization for Moore's conditions.