where P is a permutation matrix, U is upper triangular, L is lower triangular, and D is a block diagonal matrix with diagonal blocks of order 1 or 2; U and L have diagonal elements equal to 1, and have 2 by 2 unit matrices on the diagonal corresponding to the 2 by 2 blocks of D. The permutation matrix P (which represents symmetric row-and-column interchanges) and the 2 by 2 blocks in D are needed