# Difference between sparse cholesky and cholesky decomposition

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:

$$A=LDL^T$$

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:

$$PAP^T=LDL^T$$

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?