Let $\mathbf{A}\in\mathbb{C}^{n\times n}$ be a random hermitian matrix. Assume that the eigenvalues of this matrix have continuous probability distribution.
1.Can we say that the eigenvectors corresponding to these eigenvalues will also have continuous probability distribution?
2.If the answer to the first part is yes, can we say that the eigenvectors are analytic about their mean values?