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I wonder - do you know of any example where the Woodbury formula (cf. 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.

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This is quite useful is statistics. Have a look at the appendices here for example: You can find a number of other references in Google. – ivan Jun 12 '12 at 13:04
I've seen it used (and used it myself) with the Kalman filter. Matrices of that form come up when dealing with covariance matrices, particularly for normal distributions. See, for example, – Jeremy West Jun 20 '12 at 20:58

This is just a comment (but I could not find the comment button). Sherman–Morrison–Woodbury formula plays an important role in this paper

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Since there are only few answers, I'll add a shameless plug and advertise one of my papers.

Basically, we first used SMW as a computational tool to speed up a matrix inversion, and then we found out that what comes out is essentially another computational method for the same equation that was previously derived in a completely different way (Theorem 5.1). As usual, the slides might provide a better exposition than the paper.

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