If a partial pivoting strategy is observed (pivoting is restricted to row interchanges), factorization produces matrices L and U which satisfy the following equation.
P is a permutation matrix that is derived as follows:
P is initialized to I.
Each row interchange that occurs during the decomposition of A causes a corresponding row swap in P.
Recalling the definition of a linear system of equations
and premultiplying both sides by P
Using Equation 54 to substitute for PA yields
Observe that the product Pb is computed before forward substitution begins. Computationally, this implies that P can be implemented as a mapping that is applied to b before substitution.