If a complete pivoting strategy is observed (pivoting involves both row and column interchanges), factorization produces matrices L and U which satisfy the following equation.
where P is a row permutation matrix and Q is a column permutation matrix. Q is derived from column interchanges in the same way P is derived from row interchanges.
Since Equation 40 and Equation 42 are identical, P can still be implemented as a mapping that is applied to b before substitution begins. Since Equation 45 computes the product Qz after back substitution is finished, Q can be implemented as a mapping that is applied to A following the substitution process.
If A is symmetric, pivoting for numerical stability may destroy the symmetry of the LU decomposition of A. For a symmetric factorization of A, matching row and column interchanges are required. In other words, pivoting must be complete and the permutation matrices must be related as follows: