Talk:Singular value decomposition

An important extension of the singular-value decomposition theorem says that if M is a symmetric square matrix then one may take G = H, and in the case in which n=m=r (the full-rank case) and all of the singular values are different one must take G = H. That is the spectral theorem.

This is incorrect: a symmetric matrix can have negative eigenvalues, and then the spectral theorem decomposition wouldn't be a singular-value decomposition.

Something is also fishy with the term "singular vector": they are not uniquely determined by M; different singular-value decompositions of M can yield different singular vectors. What is the clean formulation of this situation? AxelBoldt 00:17 Dec 13, 2002 (UTC)

For all the textbooks and articles on matrix theory I have ever read, this is the one and only one article uses the term singular vector!!! Wshun

Well, Google gives about 3000 hits for the phrase "singular vector", and most of them seem to refer to the columns/rows of the matrices in the singular value decomposition. So the term is apparently in use, it's just that the definition is not quite clear. I restored some of the removed material about them, and also mentioned the fact that singular value deompositions always exist. AxelBoldt 19:30 Apr 6, 2003 (UTC)

Why directing the article Singular value to Singular value decomposition? It is as bizarre as directing prime number to prime number theorem, or calculus to the fundamental theorem of calculus[?]. It is better to reverse the direction. In my opinion, the two articles should not be combined together! Wshun