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Linear Equation - LU Factorization

The LU factorization is a modification method of Gauss elimination.
The linear system of equation is Ax = b
where A - n x n coefficient matrix
An LU-factorization of a given square matrix A,
A = LU
where L is lower triangular and U is upper triangular.
Example
The rows can be reordered so that the resulting matrix A has an LU-factorization for any nonsingular matrix.
Ax = b can be written as LUx = b
Ly = b where Ux = y
If main diagonal elements of Lower triangle is 1, then the method is Doolittle Method.
If main diagonal elements of Upper triangle is 1, then the method is Crout's method.
For a symmetric positive definite matrix, A
and 
So, A = LU can be written as 
The method of solving Ax = b using the above factorization is called Cholesky's method.
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