Symmetric elimination without pivoting

Positive definite matrices factor into A=LLTA=LLT (Cholesky). Symmetric indefinite matrices need a symmetric middle factor in A=LPLTA=LPLT. Then A and P have the same inertia (eigenvalues of the same sign). We construct P through elimination, so the inertias agree for all leading minors of A and P. When restricting P to be a variant of a symmetric permutation in which diagonal 1's can be replaced by 0's or −1's, it is unique.