Singular values are the square-roots of the eigen values of MM' matrix

[Subbarao Kambhampati <rao@asu.edu>]

Folks:
 In the class today, I may have given the impression that singular 
values (in the SVD decomp) are directly the eigen values of
M*M'  (or M'*M) matrix. They are actually the +ve square roots of the 
M*M' matrix  (and aren't you happy that M*M' is a square symmetric 
matrix with positive eigen values?)

The square-root makes intuitive sense since M*M' can be seen as a 
sort of "square" of M (modulo the unintended pun).

Here is the full dirt:

Given the d-t matrix, SVD is

d-t = d-f x f-f x t-d'

where

f-f are the +ve square roots of the eigen values of either d-txt-d 
(doc-doc correlation matrix)  or t-dxd-t (term-term correlation 
matrix)   (both will have the same eigen values--but different eigen vectors)

the columns of d-f are the eigen vectors of d-txt-d  (i.e. the 
doc-doc  correlation matrix)

the columns of t-f are the eigen vectors of t-dxd-t matrix (i.e. the 
term-term correlation matrix)

that is all!

Rao