Can someone please help me explain whether there exists a relationship between the LU Factor and the Cholesky Factor?
Some sources said that it holds L^T = U. However I tried to see it in an actual matrix and failed to prove it.
Lets consider this matrix:
A =
⌈2 -1 0⌉
|-1 2 -1｜
⌊0 -1 2⌋
Cholesky:
L=
⌈1.414 0 0 ⌉
|-0.707 1.225 0 ｜
⌊ 0 -0.816 1.155⌋
L^T= ⌈1.414 -0.707 0 ⌉ | 0 1.225 -0.816｜ ⌊ 0 0 1.155⌋
LU: L= ⌈ 1 0 0⌉ |-0.5 1 0｜ ⌊ 0 -0.67 1⌋
U= ⌈ 2 -1 0⌉ | 0 1.5 -1｜ ⌊ 0 0 1.33⌋
I realised that for LU factorization, the L and U matrix will differ based on my pivoting. Does the pivoting will affect the relationship between LU factors and Cholesky factors?