# 4.9 Complete Pivoting

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.

 $\mathbf{LU=PAQ}$ (57)

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.

If A and its factors are related according to Equation 57, then Equation 35 can still be solved for A by the sequential solution of two triangular systems.

 $\displaystyle\mathbf{y}$ $\displaystyle=\mathbf{Pb}$ (58) $\displaystyle\mathbf{Lc}$ $\displaystyle=\mathbf{y}$ (59) $\displaystyle\mathbf{Uz}$ $\displaystyle=\mathbf{c}$ (60) $\displaystyle\mathbf{x}$ $\displaystyle=\mathbf{Qz}$ (61)

Since Equation 56 and Equation 58 are identical, P can still be implemented as a mapping that is applied to b before substitution begins. Since Equation 61 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:

 $\mathbf{Q={P}^{T}}$