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:

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