Suppose I have a symmetric matrix $K$. Subdivide $K$ into pieces as $$K=\begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22}\end{pmatrix},$$ where $K_{21}=K_{12}^\top$. Then, the Nystrom approximation of $K$ replaces $K_{22}$ with the approximation $$K_{22}\approx K_{12}^\top K_{11}^{-1}K_{12}.$$ See e.g. this paper for an introduction.
Now, suppose $K$ is asymmetric. Here's my question: If we subdivide $K$ as above, does there exist a "Nystrom-style" approximation of $K_{22}$?
I'm OK with assuming $K_{11}$ is chosen to be square and invertible if that helps. It seems naively that I could use exactly the same formula when $K$ is not square, but of course the proof of the Nystrom approximation no longer holds; I assume an eigenvalue argument has to be replaced with an SVD.