Matrix Factorization of lower triangular and unit upper triangular.

Question

So I have a matrix.

$$A =\begin{bmatrix} 8 && -3 && 2 && -1\\ -3 && 8 && 0 && 2\\ 2 && 0 && 8 && -3\\ -1 && 2 && -3 && 8 \end{bmatrix} $$

And the $[L,U] = Doolittle(A)$ is

$$L = \begin{bmatrix} 1.0000 && 0 && 0 && 0\\ -0.3750 && 1.0000 && 0 && 0\\ 0.2500 && 0.1091 && 1.0000 && 0\\ -0.1250 && 0.2364 && -0.3946&& 1.0000 \end{bmatrix} $$ $$ U = \begin{bmatrix} 8.0000 && -3.0000 && 2.0000 && -1.0000\\ 0 && 6.8750 && 0.7500 && 1.6250\\ 0 && 0 && 7.4182 && -2.9273\\ 0 && 0 && 0 && 6.3358 \end{bmatrix}$$

The question asks to solve A

$A = L_2 \times U_2$, where $L_2$ is lower triangular and $U_2$ is unit upper triangular.

So isn't it just $U' \times L'$ to get the answer? Or do I have to use the Cholesky factorization to get the answer?

Any help would be appreciated. Thank you!

Answer 1

Not sure what your question is asking but according to the Doolittle's method A=L*U. You have found both L2 and U2.

this might also help: http://mathfaculty.fullerton.edu/mathews/n2003/CholeskyMod.html