I am confused as to the difference between the regular cholesky decomposition and the sparse cholesky decomposition (i.e. that which is performed by LAPACK's CHOLMOD for example).

I understand the Cholesky decomposition decomposes a square matrix, $A$, into two matrices $D$ and $L$ such that $D$ is diagonal, $L$ is lower triangular and:


From what I understand the Sparse Cholesky decomposition involves first working out some form of permutation $P$ matrix and applying it to a matrix $A$ in the following manor and then performing some form of Cholesky decomposition like so:


I guess my main confusion is this. Suppose I had the matrix $A$ and I have previously run a sparse cholesky decomposition on a different matrix, $B$, which had the same structure (i.e. zeros in the same entries as in $A$) and have the permutation used in that decomposition $P_B$. Can I just work out $P_BAP_B^T$ and do a regular cholesky decomposition of that to obtain the sparse cholesky decompositon of $A$ or is there more to the sparse cholesky decomposition than just finding the permutation?


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