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For an application in physics I would like to estimate the effect of a small pertubation on an ideal system. For that, I require the change to the eigenvectors and eigenvalues of an diagonal matrix with small off-diagonal pertubations. As such, we have the following matrix:

$W=\mathbf{D} + \alpha X$

where $\mathbf{D}$ is a diagonal matrix with possibly degenerate eigenvalues, $\alpha$ is a scalar << 1 and $X$ is a hermitian complex matrix where all the entries are smaller than the smallest value of $D$, with 0 on the diagonal. Is there any way to compute the changes to the eigenvalues and eigenvectors of $W$ as a function of $\alpha$? Even if it is only an approximation it would be a big help. Normally I would use first order perturbation theory to solve this problem, but due to the nature of the pertubation all of these components are zero. As there are degeneracies involved, computing higher order perturbation coefficients is possible but it would be demanding. The only thing that is changing is the strength of $\alpha$, so I hope that it should be possible to compute the change, or a good approximation, analytically.

I have heard that there exists such a thing as a derivative of the singular value decomposition, but I don't quite get how to use that in this case. I also had a look at https://en.wikipedia.org/wiki/Eigenvalue_perturbation#Results, and I think that what I would require is something like a derivative with respect to $\delta$.

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    $\begingroup$ is $X$ Hermitian? (you say "symmetric complex", do you mean Hermitian?); and why does first order perturbation theory give zero? $\endgroup$
    – Carlo Beenakker
    Commented Sep 27, 2019 at 11:00
  • $\begingroup$ Thanks for pointing that out. Yes, $X$ is hermitian. First order pertubation theory gives zero as all the entries on the diagonal of $X$ are zero. I updated the question to include that. $\endgroup$
    – Dirk B.
    Commented Sep 27, 2019 at 11:14
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    $\begingroup$ Eigenvalues and singular values are not the same thing. Although for hermitian matrices the latter are the absolute values of the former. Anyway it seems that your problem can be solved with standard (degenerate) perturbation theory. $\endgroup$
    – lcv
    Commented Sep 27, 2019 at 11:23
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    $\begingroup$ you just want to do degenerate second order perturbation theory; any quantum mechanics textbook will explain how, you basically first want to diagonalise $W$ in the subspace of the degenerate eigenvalues of $D$; with a specific information on how $X$ looks like there is not much more that one can say here. $\endgroup$
    – Carlo Beenakker
    Commented Sep 27, 2019 at 13:14

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