I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows:
Let $A$ be random matrix s.t. $A=(a_{ij})$ and $a_{ij}\sim N(0,\sigma^{2})$. Let $\lambda$ be eigenvalue of random matrix A and let $x$ be eigenvector of $A$. As \begin{equation} Ax=\lambda x \end{equation} we can expand this as \begin{equation} a_{i1}x_{1}+\cdots+a_{in}x_{n}=\lambda x_{i} \end{equation} And IF WE ALLOW $x_{i}$ to be scalars not depending on random variables $a_{ij}$, \begin{align} \mathbb{E}(a_{i1}x_{1}+\cdots+a_{in}x_{n})&=\mathbb{E}(\lambda x_{i})\\ \mathbb{E}(a_{i1})x_{1}+\cdots+ \mathbb{E}(a_{in})x_{1}&=\mathbb{E}(\lambda)x_{i}\\ 0&=\mathbb{E}(\lambda)x_{i} \end{align} Thus, $\mathbb{E}(\lambda)=0$(assuming $x_{i}\neq0$) And by same method, we can show variance of $\lambda$ to be $\mathbb{V}(\lambda)=n\sigma^{2}$.
I think there is a serious problem in that I assumed $x_{i}$s to be scalar.
Is there any way to solve this problem?
Is there any depiction or theorem about eigenvalues of a random matrix defined as I defined?
The matrix model you described is called the real Ginibre ensemble, provided that all of your $a_{i,j}$'s are i.i.d. The result is not standard, which can be found from this paper.