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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?

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  • $\begingroup$ How are the eigenvectors distributed? Once you pick some constellation, you can choose the eigenvectors arbitrarily in the sense that choosing the eigenvalues corresponds to selecting a diagonal $\Lambda$ and then choosing the eigenvectors is tantamount to selecting some unitary $U$ to get $A=U \Lambda U^*$. $\endgroup$ – copper.hat Mar 20 '15 at 1:27
  • $\begingroup$ I am using moment method to reconstruct pdf. So I am using the first two moments of eigenvectors to approximate its density as complex normal. Although normal distributions are continuous, but since I am using only first two moments, I am not sure of the higher moments. So, I wanted to see if there is any theory to prove that the eigenvectors will be continuous. $\endgroup$ – user146290 Mar 20 '15 at 1:31

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