Let $M$ be a real symmetric matrix of size $N$ with its components $M_{ij}$ following a normal distribution centered around 0.

Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\in\mathbb{R}$: $$\sum_j M_{ij}x_j=\lambda x_i$$

I know that in that case the eigenvectors of different eigenvalues are independent from one another. However:

- Is there any universality property regarding the distribution of the elements of the eigenvectors? If we want normalised eigenvectors I suppose that each entry of the eigenvector must have a variance that scales with $1/N$?
- Is there any known relation between the matrix elements and the corresponding eigenvector elements? I am particularly interested in the correlation $\langle M_{ij}x_j\rangle$?