The LU factorization is perhaps the most common method for solving linear systems of equations. It is equivalent to Gaussian elimination, but preserves information so that new systems with the same matrix can be solved efficiently. This factorization has the form
where is lower triangular (that is, if ), and is upper triangular (that is if ).
Usually it is used in a modified form where is a permutation matrix (that is, all entries are zero or one, and only one entry in each row and in each column is non-zero).
To compute the LU factorization use recursion, which is essentially the algorithmic version of induction.
The case is easy: .
then we can recursively compute the LU factorization of . Then an LU factorization of can be given by
where , , and , which can all be solved provided is non-singular.
This method can be modified to give a constructive proof of the existence of the LU factorization provided every principal (north-west) square submatrix of is non-singular. This recursive construction creates an matrix with 1's on the diagonal. It can be easily modified to create a matrix with 1's on the diagonal.
A different recursive construction can be given where we use the decomposition
This gives equivalent computations, but done in a different order.