Studying proof of $QR$ algorithm for finding eigenvalues, I was faced with fact: $\Sigma^k L \Sigma^{-k} \rightarrow L^{'}$, where $L^{'} = {\rm diag}(L)$, $L$ is lower triangle matrix, $\Sigma$ - diagonal matrix.
I have no idea, how to prove this fact, and I want some intuition behind the formal proof.
I must add that matrix $\Sigma - $ matrix of eigenvalues, sorted by value, in ascending order. I think then I get intuition of this equation. The biggest eigenvalues from $\Sigma^{-k}$ have more contribute at non-diagonal elements, so they decreasing as the degree increases.