International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 8, Pages 505-509

On the relation between Moore's and Penrose's conditions

Gaoxiong Gan

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.


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.