# Cholesky of a submatrix

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.

• From a programmatic standpoint, this isn't possible without looping through the entire Cholesky roll. This is because each coefficient in a Cholesky decomposition depends on all values above it and to it's left in A and B, so if A is not identical to B, you won't get an "overlapping" or in any way related Cholesky factorization. I worked on this as part of research into active-set NNLS, and was unable to find any way to project or transform a pre-conditioned Cholesky decomposition L (from A) into M given B, where B is a symmetrical subset of A. Mar 29, 2021 at 13:51