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Let $A$ be a real, symmetric $n \times n$ matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ where at least two of the eigenvalues are zero. Let $V$ be a real, symmetric $n \times n$ matrix. Consider $A_\varepsilon = A + \varepsilon V$, where $\varepsilon$ is a small number. I am looking to determine the change in the eigenvalues induced by the perturbation $\varepsilon V$.

My question is very similar to this post, except that in my case, I have repeated zero eigenvalues. I have been searching the literature for a method that would work in this case, but I haven't been able to find anything.

Dawson Beatty
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    It seems that the accepted answer in the post you linked answers your question, right? – NN2 Sep 27 '23 at 15:00
  • Unfortunately no, I tested the accepted answer but it gives the wrong solution when the eigenvalue is zero. The problem with the given solution (I think) is that when implemented numerically, the eigenvalues will not exactly be zero, and so the eigenvectors are orthogonal but mostly determined by numerical errors. The method fails because the eigenvectors are nonsense. – Dawson Beatty Sep 27 '23 at 15:05

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Let $\lambda$ be an eigenvalue of $A$ and $P$ the projection onto the associated eigenspace. If $P VP $ has distinct eigenvalues $\tilde{\lambda}_1, \dots ,\tilde{\lambda}_m$ with each eigenspace being one dimensional, then $\lambda$ splits into $m$ eigenvalues $\lambda_1 (\varepsilon) , \dots , \lambda_m(\varepsilon)$ of $A_\varepsilon $ which are given by $$ \lambda_i (\varepsilon) = \lambda +\varepsilon \tilde{\lambda}_i +o(\varepsilon).$$

For a derivation and more general cases, see Chapter 2 (in particular §2 section 3 "the reduction process") in the book "Pertubation Theory For Linear Operators" by Tosio Kato.

jd27
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