I have a large-ish matrix that is the Kronecker product of two smaller matrices $V = A \otimes B$. A and B are both positive definite. I want to take the Cholesky decomposition of arbitrary subsets of V. By "arbitrary subsets" I mean I want to be able to compute the Cholesky decomposition of $V_\gamma$, where $\gamma$ is an arbitrarily chosen set of indices, and $V_\gamma$ is the matrix formed by taking those rows and columns from $V$ that are contained in $\gamma$.
If $A = LL'$ and $B = MM'$ then $V = (L \otimes M)(L\otimes M)'$. Can I use $L$ and $M$ to easily compute the Cholesky decomposition of $V_\gamma$?
If I can't compute the Cholesky, but could somehow compute the determinant that would be almost (but not quite) as good.