I wonder - do you know of any example where the Woodbury formula (cf. http://en.wikipedia.org/wiki/Woodbury_matrix_identity) was crucially used to prove anything?
It might be a useful computational procedure but I've tried many times to use it in proofs and always ended up exasperated - it just replaces a nasty-looking matrix expression with an even nastier-looking expression.
But I've not given up hope on it and therefore am asking for examples of successful theoretical usage.
P.S. The special rank one case (aka the Sherman-Morrison formula) is of course mighty useful. I am asking specifically about the general case.