LDL factorization

Every nonsingular symmetric matrix $$A$$ can be factored as

$$A=PLDL^TP^T$$

with $$P$$ a permutation matrix, $$L$$ lower triangular, $$D$$ block diagonal with $$1\times 1$$ or $$2 \times 2$$ diagonal blocks. The cost of this factorization is $$(1/3)n^3$$.