eigenvalues of many dense symmetric real matrix that are 'close' to each other

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Henry Wolkowicz 2019-6-15

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https://ww2.mathworks.cn/matlabcentral/answers/467273-eigenvalues-of-many-dense-symmetric-real-matrix-that-are-close-to-each-other

编辑： David Goodmanson 2019-6-16

I have to find the eigenvalues of many dense symmetric real matrix that are 'close' to each other, i.e. they are not much different. Can I speed up eig or some other code if I know the spectral decomposition of A and want to find it for a nearby B. I.e. I have A = UDU' as the spectral decomposition and want to find it for B where

B-A is small. I know this can be done for eigs with 'restarts'. But what about finding all the eigenvalues with eig?

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David Goodmanson 2019-6-16

编辑：David Goodmanson 2019-6-16

Hi Henry,

If the eigenvalues are not too closely spaced (no repeated ones either) then a simple first order approximation gives a quick look at how much the eigenvalues change. Let A1 = B-A. The diagonal elements of

E1 = U'*A1*U

are the shifts in the eigenvalues, to first order. Perturbation theory can provide results for higher orders, using increasingly complicated expressions.

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