Let $v_i\in\mathbb{R}^n$ such that each component is bounded by $|v^{(k)}_i|\leq\frac{1}{\sqrt{n}}$. Consider the matrix
$$ A=\frac{1}{m}\sum_{i=1}^mv_i v^T_i \in\mathbb{R}^{n\times n} $$ and note that A is obvious symmetric and has eigenvalues in $[0,1]$. Let $(\lambda_j, w_j)$ be the eigenvalues and corresponding eigenvectors. Is it possible to characterize $w_i$ in terms of $v_i$? For $m=1$ i know that $A$ has rank $=1$ so all eigenvalues are equal zero exept one which is equal $v^T_1v_1$ with eigenvector equal $v_1$ but how about $m>1$ ?
