- Author
- G. W. Stewart
- Published
- December 1993
- Kind
- Historical survey article in a mathematics journal (SIAM Review)
- Original
- www.math.ucdavis.edu/~saito/courses/229A/stewart-svd.pdf
- Subject
- Mathematics and optimization
What it is
G. W. Stewart wrote this survey for SIAM Review. It sets out to credit the people who established the decomposition and its theory. It begins with background on older matrix factorizations by Gauss and others.
What it says
- Stewart surveys five mathematicians who built the theory: Beltrami, Jordan, Sylvester, Schmidt and Weyl.
- Beltrami's 1873 derivation covered only square nonsingular matrices with distinct singular values.
- Jordan found the decomposition independently a year later and avoided degeneracies by deflation.
- Schmidt's 1907 work extended it to integral equations and gave optimal low-rank approximations.
- Weyl gave a general perturbation theory and a cleaner proof of Schmidt's theorem.
- Stewart says Schmidt's approximation theorem is often wrongly called the Eckart-Young theorem.
- Stewart says the name singular value seems to come from integral equations.
What evidence it gives
The paper paraphrases the original derivations and quotes some passages directly. It cites a reference list of dated primary papers. Stewart also comments on how clear each derivation is.
Interests and limits
The scanned PDF was read through text recognition, which garbled most equations. The paper is a 1993 survey, and it sketches later work only briefly. Stewart says each sketch omits detail and urges readers to consult the sources. It offers argument and paraphrase, not new results.