Updating eigen-decomposition of symmetric matrix $A$ to eigendecomposition of $A+D$ where $D$ is low-rank diagonal

Question

Given a symmetric positive definite matrix $A$ and a mostly-zeros non-negative diagonal matrix $D$, is there a way to cheaply update the eigenvalues and/or eigenvectors of $A$ to that of $A+D$? Ideally I'm looking for something akin to the Woodbury matrix identity.

Answer 1

I would recommend reading http://www.unige.ch/~gander/consulting/X/EigenUpdate.ps.gz and having a look at the cited work of Golub and Van Loan. They show howto update matrices with rank-one-changes. You can understand you update matrix $D \in\mathbb{R}^{n\times n}$ as a sum of $n$ rank-one-updates. Good luck!