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

Question

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

Answer 1

• 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 $\mathbb{E}[\lambda_k]=0$.