Let $M$ be a real random $d\times d$ symetric matrix. I know that since $M$ is symmteric we can always have an Eigen-Decompsotion in its Diagonal Form.
Suppose the Eigen-Vectors are such that $V_1,\cdots,V_d$ are uniformly distributed over the $d$ dimensional sphere.
Can we say anything about eigenvalues of $M$? Can we say that the eigenvalues would be close to each other?