I haven't seen the article ... was this post a summary of the contents? Whether or not, I don't understand the connection between V,W and P,Q here. The geometric interpretation as it stands is at best incomplete, at worst self-evidently fallacious: singular values of a general matrix M may exceed unity, yielding complex angles. WFL On 12/22/12, Dan Asimov <dasimov@earthlink.net> wrote:
Anyone else see the article on the SVD in the current monthly?
If so, I'm curious what you think of it.
--Dan
((( The basic SVD theorem says that any KxL real matrix M can be expressed uniquely as
M = P D Q
where P (KxK) and Q (LxL) are orthogonal matrices, and D (KxL) is diagonal with D_11 >= D_22 >= . . . D_pp >= 0, where p = min{K,L}.
OK, the "singular values" D_jj are unique. P and Q are unique only if the D_jj are all distinct.
This has a geometric meaning: For any two subspaces V, W of some R^n, with dim(V) = K, dim(W) = L, then K and L have p = min{K,L} "principal angles" 0 <= theta_1 <= theta_2 <= . . . <= theta_p <= pi/2, with D_jj = cos(theta_j) for all j = 1,...,p, such that:
(*) the theta_j's completely determine the relative positions of K and L in R^n.
(Here theta_1 is the least angle between any v in V and w in W. Now take the orthogonal complements of v in V and of w in W: theta_2 is the least angle between vectors in these subspaces. Lather, rinse, repeat.)
(*) means that for any other subspaces V', W' of R^n with dim(V') = dim(V),
dim(W') = dim(W), and having the same principal angles, there is an isometry h: R^n -> R^n in O(n) with h(V) = V' and h(W) = W'.
For n >= K+L, any sequence of principal angles 0 <= theta_1 <= theta_2 <= . . . <= theta_p <= pi/2 can arise. )))
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