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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.

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    $\begingroup$ 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. $\endgroup$
    – zdebruine
    Mar 29, 2021 at 13:51

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