From: "Urban Lundman" <urblun-6@sm.luth.se>
Subject: Woodbury Formula for a special case
Date: Wed, 25 Jul 2001 15:50:15 +0200
Newsgroups: sci.math.num-analysis
Summary: Sherman-Morrison-Woodbury formula for updating inverses

Hi,

I wonder if there is any way to simplify the Woodbury (or
Sherman-Morrison-Woodbury) formula for this special case:

    inv(C + w'*w) = ...,

where C is symmetric and w is a matrix of suitable size (but not a single
row). C is a covariance matrix (C = inv(A'*A) for some matrix A).
(w' denotes the transpose of w)

The formula states that

    inv(A + U*D*V') = inv(A) - inv(A)*U*inv[inv(D) + V'*inv(A)*U]*V'*inv(A)

if hope I got it right.

I've tried some myself, ending up with

    C*E*C - W*E*W = C - W,

(again if I got it right) where E is the expression I want so simplify (that
is E = inv(C + w'*w)) and W = w'*w, but I cannot get an explicit, "simple"
expression for E.

Thanks in advance,

    Urban