L U Decomposition method : In these methods the coefficient matrix A of the given system of equatiron AX = b  is written as a product of a Lower triangulat matrix L and an Upper trigular matrix U, such that A = LU where the elements of L = (l_ij = 0 for i < j) and the elements of U = (u_ij = 0 for i > j) that is, the matrices  L and U look like

Now using the rules of matrix multiplication

This gives a system of n² equations for the (n² + 1) unknowns(the non-zero elements in L and U). To make the number of unknowns and the number of equations equal one can fix the diagonal element either in L or in U as '1' then solve the n² equations for the remaining n² unknowns in L and U. This can be written as

Once the matrix A is --- into LU form we can write the system as LUx = b
if Ux = y Þ Ly = b
solve y from Ly = b and x from Ux = y.
Since L and U are respectively lower diagonal and upper diagonal in structure we can use forward subsititution and back substitution to find y and x from the respective equations.

Example:
 
 

Cholesky decomposition : The LU decomposition defined in the previous section can be simplified if the coefficient matrix A in the given system of equations Ax = b is symmetric and positive definite. In this case the decomposition can be modified as A = LL^T so that it is sifficient to compute the lower triangular matrix L and the upper triangular matrix can be obtained by taking the transpose of L that U = LT. Once A is decomposed into LU(Here LL^T) the same iterative scheme used in LU decomposition method can be used here to compute the unknown vector X. That by solving two systems Uy = b first and then Lx = y.