I am trying to find the $LDL^{T}$ factorization of the following matrix $$ A = \begin{bmatrix} 1 & b \\ b & 4 \end{bmatrix} $$ when $b$ is in the range of positive definiteness.
I have already determined that $b$ is in the range of positive definiteness when $b < 2$. However, I don't really understand how to find the $LDL^{T}$ factorization. I know how to do it with a matrix with all numbers, but when I applied the same strategy to this particular matrix I got a big mess when I calculated $LDL^{T}$.
The only thing I can think to do is plug in a value less than 2 and do it the way I normally do it, but I have a feeling any time I change the value for $b$ I will get a different $LDL^{T}$ factorization.
Can anyone give me any advice?
$\endgroup$ 51 Answer
$\begingroup$Apply one step of the Cholesky decomposition algorithm to get:
$\endgroup$$A = \begin{bmatrix} 1 & 0 \\ b & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 4-b^2 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ b & 1 \end{bmatrix}^T$.
