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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: http://mathoverflow.net/questions/34252/eigenvalues-of-ab-where-a-is-symmetric-positive-definite-and-b-is-diagonal2011-02-05
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    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?2011-02-05
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    hardmath: Yes, $D$ is a diagonal matrix and even its diagonal is just mostly zeros.2011-02-06

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