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In a previous question I asked about the stability of eigenvalues with respect to diagonal perturbations. Following results from the book Matrix Analysis (by Roger A. Horn & Charles R. Johnson) the results are quite nice for symmetric matrices. However, they explicitly say that this is not the case for eigenvectors and they give the counter example $\begin{pmatrix} 1& \varepsilon \\ \varepsilon & 1\end{pmatrix}$.

In what I have in mind, only diagonal perturbations are involved, so the counterexample given above is not relevant. I have the following question:

Suppose $A$ is a real, symmetric, positive-definite matrix and $D$ be a diagonal matrix of the same size. Let $\lambda_1$ be the smallest eigenvalue of $A$ and assume that it is simple.

  1. Is there a simple counter example to the stability of eigenvectors in this case? (The eigenvectors corresponding to the smallest eigenvalue of $A+tD$ do not converge towards the eigenvectors of $A$, associated to $\lambda_1$ as $t\to 0$.)

  2. If there is no counterexample, do you have a reference for such a stability result?

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  • $\begingroup$ It seems this is answered in your link, specifically the quote from Lax's book, Theorem 8, pg 130. Or am I misunderstanding your question? $\endgroup$
    – user101142
    Apr 5, 2018 at 14:01
  • $\begingroup$ The general topic of continuous dependence of eigenvectors on the matrix is discussed in this previous question: MO116123. One of the answers is again by Peter Michor, but with more details. $\endgroup$ Apr 5, 2018 at 15:39

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Simple eigenvalues depend smoothly on the matrix by the implicit function theorem applied to the characteristic polynomial (parameterized by the symmetric matrix). For the general situation see this paper and references therein.

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