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.
Cost: 
flops.
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.
