Subject: Re: Moore-Penrose Inverse
Date: Fri, 12 Nov 2000 18:54:54 -0500
From: Alex Bogomolny
Here's an answer to your question. It came by the way of a friend of mine, Florian Potra, currently at the University of Maryland. Florian went directly to the source: M.Z. Nashed who edited the book
M. Z . Nashed, editor, Generalized Inverses and Applications, Academic Press, 1976.
All the best,
- E.H. Moore in 1920 generalized the notion of "inverse" of a matrix to include all matrices. Moore's definition of the generalized inverse of a (rectangular) matrix is equivalent to the existence of a matrix G such that AG and GA are appropriate projectors. Moore was a professor at the University of Chicago and was one of the leading American analysts in the early 1900s. Much of his work remained dormant for a long time since he wrote in an obscure way.
- Unaware of Moore's work, Penrose showed in 1955 that there exists a unique matrix B such that ABA = A, BAB = B, AB and BA are Hermitian. In 1956 he published a paper showing that the genaralized inverse he has defined has the least squares property. Moore did not have this. Penrose's only 2 papers on generalized inverses are short and elegant, and are based on his Ph.D. at Sheffield, UK.
- Penrose is the well known ROGER PENROSE who has profound contributions to tiling, mathematical physics, and other areas. I recall that he gave the Gibbs' lecture at the AMS a few years ago.
- Neither Moore nor Penrose considered generalized inverses of operators on Hilbert space, but Tseng, a Chinese student of Moore and Barnard, considered this in his Ph.D. dissertation at Chicago, 1933.