I’m pushing to submit a preprint on the Nystrom method that has been knocking around for the longest time.

I find myself running into problems centering around expressions of the type \(B^{-1}A\), where \(A, B\) are SPSD matrices satisfying \(B \preceq A\). This expression will be familiar to numerical linear algebraists: there \(B\) would be a preconditioner for a linear system \(A x = b,\) and the relevant quantity of interest is the spectral radius of \(B^{-1} A\).

It’s not hard to show that the spectral radius of this product is at most 1… but instead, I’m interested in the norm of this product. Because the spectral radius of the product is at most 1, we can use the bound

\[

\|B^{-1} A\|_2 \leq \kappa(U_{B^{-1}A})

\]

where \(\kappa(U_{B^{-1}A})\) is the two-condition number of the matrix of eigenvectors of \(B^{-1}A \).

In the applications I’m interested in, some rough numerical experiments show that this bound is good enough for my purposes (the two terms are of the same order). Assuming this is the case, how can I get a good theoretical bound on this condition number?