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.


This problem seems to be called rank-one downdating of the Cholesky decomposition. Here is a technical report discussing the technique


More references are given in the first answer to this post


  • $\begingroup$ I want the way to represent it as an explicit form, but anyway thank you for answering. $\endgroup$ – fiverules Apr 13 '14 at 16:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.