# Calculating Cholesky decomposition directly via Cholesky of submatrix

I want to represent a Cholesky decomposition of $(n+1) \times (n+1)$ matrix $B$ which is of the form

$B = \begin{bmatrix} k& v^T \\ v&A \end{bmatrix}$

where $k>0$, $v \in \mathbb{R}^{n}$ and $A$ is $n \times n$ positive definite matrix.

If the Cholesky decomposition of $A$ is given by $A = LL^{T}$ where $L$ is a $n \times n$ lower triangular matrix, how can I represent the Cholesky decomposition of $B=MM^{T}$ in terms of $k$, $v$ and $L$?

Here is my progress:

If $M = \begin{bmatrix} a& 0 \\ u&C \end{bmatrix}$, then since we have

$\begin{bmatrix} a& 0 \\ u&C \end{bmatrix} \begin{bmatrix} a& u^T \\ 0&C^T \end{bmatrix} = \begin{bmatrix} k& v^{T} \\ v&A \end{bmatrix}$,

we get $k=a^{2}$, $v=au$, $uu^{T}+CC^{T} = LL^{T}$ as $A=LL^{T}$.

Hence $a = \sqrt{k}$, $u = v/\sqrt{k}$, therefore, $a$ and $u$ can be represented in terms of $k$ and $v$.

However, I'm stuck on representing $C$ in terms of $k$, $v$ and $L$. Although I have $CC^{T}=LL^{T}-uu^{T}$, I don't know how to start from here.

I know that each entry $c_{ij}$ of $C$ can be computed by an algorithm of Cholesky decomposition, but I want to represent $C$ itself in terms of $k$, $v$ and $L$.

How can I do this? Thank you.