I have a personal example, with by Matrices; Theory and Applications (GTM 216, Springer-Verlag, 2000). A couple of years ago, I found a proof of almost sure convergence of the Jacobi method for computing the spectrum of a Hermitian matrix, when one uses the random strategy. I was not sure of the novelty of it (could anyone comfort me confirm it?), and I just included it in the second edition, which is going to appear in a month or two.