Distribution of eigenvectors of random matrices and link with the components of the matrix

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 $$\langle M_{ij}x_j\rangle$$?

• 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 $$\mathbb{E}[\lambda_k]=0$$.
• (1) The first point is the Porter-Thomas distribution of the Gaussian Orthogonal Ensemble (GOE), it holds only for $N\gg 1$ and only if you are considering the joint distribution of a small subset $n\ll N$ of the elements. (2) The second point holds exactly for any $N$ in the GOE, so if the elements of $M$ are uncorrelated; if they are correlated, it holds for $N\gg 1$ because the GOE is reached in that limit. Mar 16 at 7:36