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 $\lambda_1$)

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

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?