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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.

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Here is a similar question:… –  PEV Feb 5 '11 at 22:34
When describing diagonal matrix $D$ as "mostly-zeros", do you mean this is true of the diagonal entries? E.g. a single nonzero entry on the diagonal would be of interest? –  hardmath Feb 5 '11 at 23:34
hardmath: Yes, $D$ is a diagonal matrix and even its diagonal is just mostly zeros. –  user6681 Feb 6 '11 at 0:40

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I would recommend reading 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 your update matrix $D \in\mathbb{R}^{n\times n}$ as a sum of $n$ rank-one-updates. Good luck!

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