TITLE: Efficient rank-two updates of an eigenvalue decomposition (or more generally SVD)
QUESTION [14 upvotes]: Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. Golub, et al.1 and Bunch, et al.2 have shown that given such an $A$, the eigenvalue decomposition of $A+\rho xx^t$ may be computed efficiently.
Has anything similar been done for the case where the update is of the form $A+B$, where $B=uv^t+vu^t$ is a rank-two symmetric matrix (note we can't just do two rank-one symmetric updates)?
What if we relaxed the insistence that $B$ be symmetric and asked instead for an efficient computation of the SVD of the update $A+B$?
1Golub, Gene H., Some modified matrix eigenvalue problems, SIAM Rev. DOI: 10.1137/1015032, JSTOR, ZBL0227.65025.
2Bunch, J.R., Nielsen, C.P. & Sorensen, D.C. Rank-one modification of the symmetric eigenproblem. Numer. Math. 31, 31–48 (1978). DOI:10.1007/BF01396012, eudml, ZBL0369.65007

REPLY [9 votes]: Just wanted to note that if one knows that the rank two update is of the form $uv^t + vu^t$ and one knows $v$ and $u$ then it is easy to find $x,y$ such that $uv^t + vu^t = xx^t - yy^t$. 
$x,y = \sqrt{\frac{|v|}{2|u|}} (u \pm \frac{|u|}{|v|}v)$. Once we know $x,y$ then the problem reduces to two symmetric rank one updates and we can use the symmetric rank one update methods  cited in the question.
proof : $ xx^t - yy^t  = \frac{|v|}{2|u|}( u + \frac{|u|}{|v|}v)(u + \frac{|u|}{|v|}v)^t - \frac{|v|}{2|u|}( u - \frac{|u|}{|v|}v)(u - \frac{|u|}{|v|}v)^t  = \frac{|v|}{2|u|}(\frac{|u|}{|v|} (2uv^t + 2vu^t)) = uv^t + vu^t$
I haven't checked if doing so would be more efficient than the more general method of Brand but I suspect that it would be.