# update SVD of linear combination after changing scalar weight

There is a lot of work done on low rank updates to inverses or SVDs (or similar decompositions), but I am wondering what can be said about the inverse or SVD

$(A + c \times B)$

in terms of $(A + d \times B)$ for some other $d$. That is, in general full-rank updates, but along very particular dimensions.

$c$ and $d$ are positive scalars, and $A$ and $B$ are symmetric, positive definite matrices. Quantities like this arise in MLE-like computations in linear mixed models (e.g. http://onlinelibrary.wiley.com/doi/10.1111/1467-9868.00092/abstract).

I don't think there is any computational advantage in general. For any pair of symmetric $M$, $N$ and given $c\neq d$, one can find symmetric $A, B$ such that $M=A+cB$, $N=A+dB$, so essentially you are asking "if I already know the eigendecomposition of a symmetric matrix $M$, can I get the one of any other completely unrelated symmetric matrix $N$ for cheaper?"
In your question $A$ and $B$ are constrained to be positive definite, but that restriction doesn't matter because one can always add multiples of the identity, changing the eigendecomposition in a predictable way.
If you want a decomposition that makes sense for the whole matrix pencil $A+\lambda B$, you can use a QZ decomposition $A=QTZ, B=QSZ$, with $T$ and $S$ upper triangular. Those $Q$ and $Z$ are two different orthogonal matrices, though.