Find the $LDL^{T}$ factorization of $A$ when in the range of the positive definite

Question

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?

Answer 1

Apply one step of the Cholesky decomposition algorithm to get:

$A = \begin{bmatrix} 1 & 0 \\ b & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 4-b^2 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ b & 1 \end{bmatrix}^T$.