Modules
The full pivoting LU decomposition is such that:
$$PAQ^\top = LU,$$
where $P$ and $Q$ are permutation matrices, $L$ is a unit lower triangular matrix, and $U$ is
an upper triangular matrix.
The partial pivoting LU decomposition is such that:
$$PA = LU,$$
where $P$ is a permutation matrix, $L$ is a unit lower triangular matrix, and $U$ is
an upper triangular matrix.