Suppose we have a random vector ${\bf X} \in \mathbb{R}^n$ where every element of ${\bf X}$ has a per element bound $ |X_i| \le a_i$.
Now let ${\bf K}_{\bf X}$ be the covariance matrix of ${\bf X}$.
My question: Can we give bounds on the individual eigenvalues of ${\bf K}_{\bf X}$. Recall, that ${\bf K}_{\bf X}$ is symmetric and positive semi-definite.
I was able to give a bound on the sum of eigenvalues (here we use the fact that sum of eigen values equal to the trace of ${\bf K}_{\bf X}$) \begin{align} \sum_{i=1}^n \lambda_i = Tr({\bf K}_{\bf X}) = \sum_{i=1}^n E[X_i^2] \le \sum_{i=1}^n a_i^2. \end{align}
However, I am not sure how to do it or if it can be done for the individual eigenvalue.