Suppose that $A$ is an $n\times n$ diagonal matrix with positive diagonal elements and $\Pi$ is a random $k\times n$ matrix that could be (a) i.i.d. Gaussian, or (b) $k$ rows of a random orthogonal matrix
The matrix I am considering is $X = \mathbb{E}\{\Pi^T (\Pi A^{-1}\Pi^T )^{-1} \Pi\}$. I know that $\mathbb{E}\{\Pi A^{-1}\Pi^T\} = \frac{\|A^{-1}\|_1}{n} I$ but $\Pi$ is correlated with what gets multiplied outside the inverse. So I have no good idea how to analyse the eigenvalues of $X$. Numerical experiments suggests that large eigenvalues of $X$ are about the same. Any explanations for this?
Thanks.