GE computes an factorization of , where is unit lower triangular and is upper triangular.
Theorem: There exists a unique factorization of if and only if is non-singular for . If is singular for some then the factorization may exist, but if so it is not unique.
Proof: By induction (from Golub and Van Loan, 1992).
The effect of partial or complete pivoting is equivalent to multiplying the matrix by a permuation matrix . GE with partial pivoting (GEPP) applied to is equivalent to GE without pivoting applied to the row-permuted matrix . This holds for all n: GEPP computes a factorization .
Computing an factorization is equivalent to solving the equations:
If these non-linear equations are examined in the right order, they can be easily solved. For added generality let , , and consider an factorization with and ( is lower triangular). Suppose we know the first k - 1 columns of and the first k - 1 rows of . Setting ,
We can solve for the underlined elements in the kth row of and then the kth column of . This process is called Doolittle's method. It computes the factorization, assuming that the factorization exists.
Doolittle's method is mathematically equivalent to GE without pivoting, for we have,
Doolittle's method is well suited to calculations by hand or with a desk calculator, because it obviates the need to store the intermediate quantities . It is also attractive when we can accumulate inner products in extended precision. It is straightforward to incorporate partial pivoting into Doolittle's method. However, complete pivoting cannot be incorporated without changing the method.