LU Decomposition vs. Cholesky Decomposition What is the difference between LU Decomposition and Cholesky Decomposition about using these methods to solving linear equation systems?
Could you explain the difference with a simple example?
Also could you explain the differences between these decomposition methods in:


*

*inverse of a matrix

*forward and backward substitution

*pivoting


Both LU and Cholesky Decomposition is matrices factorization method we use for non-singular( matrices that have inverse) matrices. In general basic different between two method. the later one uses only for square matrices (A = A^T). however LU decomposition we can use any matrices that have inverses. for example see the following equation with 3 unknown 
2x + y 3z = 4
2x - 2y -z = -1
-2 + 4y z = 1
the above equation since the coefficient of a variable can't form a square matrix so we use LU decomposition to factorize the matrices, basically two steps process first row reductions until make all zero below the main diagonal so that we find upper matrix (U) and save all the factors we use on this step to substitute to lower matrix(L) and put the values of 1's to the main diagonal and 0s to above the main diagonal. using either of two (L,U)  we can easier solve the equation through back substitution values of x, y & z are 3/2, 1, 0. 
Cholesky factorization simple example Matrix A;
2x + y = 2
x + 2y = 4
this method use A = LL^T. Simply taking 2x2 lower triangular matrix multiply(components) 
with its transpose (with variables values).and matches with the coefficient of Matrix A and try to solve the unknown variable so that you can factor A with L, L^T.
