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 $\langle M\_{ij}x\_j\rangle$?



Distribution of eigenvectors of random matrices and link with the components of the matrix
• For large $N$ the elements of an eigenvector have independent Gaussian distributions with zero mean and variance $1/N$.


• To find $\mathbb{E}[M\_{ij}x\_j]$ I decompose the matrix $M$ into eigenvalues and eigenvectors,
$$M\_{ij}=\sum\_{k} O\_{ki}\lambda\_k O\_{kj},$$
with an orthogonal matrix $O$. Then
$$\mathbb{E}[M\_{ij}x\_j]=\sum\_{k} \mathbb{E}[O\_{ki}\lambda\_k O\_{kj}O\_{1j}].$$
This vanishes because eigenvectors and eigenvalues are independent, and $\mathbb{E}[\lambda\_k]=0$.


