Re: question on the svd

From: Jentje Goslinga (
Date: 12/16/04

Date: Thu, 16 Dec 2004 07:00:45 GMT

rm wrote:
> I read that for a non-square matrix, A, eg: mxn, m<n only the
> singular values D are unique. A = U*D*V'
> And that A can be expressed as: A = U1*D*V1' = U2*D*V2'
> How does one compute different possibilities of the left and right
> singular matrices ie the set of {U} and the set of {D}? and the
> number of solutions that exist in the set?

This is not so mysterious, you can work minus signs into
both the left and right matrices by writing

D = Ik.D.Ik

with Ik a Unit Matrix with its k-th diagonal element negated.
Now left multiply Ik into V' and right multiply into U.
Work out for yourself in how many ways this can be done.

This is not really any different from the symmetric eigenvalue
problem where eigenvectors are also determined up to a
multiplication by -1.

The folks in sci.math.num-analysis probably can say it better.

> Thanks!

Jentje Goslinga