$B_{(n+1)(n+1)}$ = $ \begin{bmatrix} A & u \\ u^T & 1 \\ \end{bmatrix}$ = $\begin{bmatrix} L_{11} & 0 \\ L_{21} & l_{22} \\ \end{bmatrix} $ $\begin{bmatrix} L_{11}^T & L_{21}^T \\ 0 & l_{22} \\ \end{bmatrix} $
Here A is nxn matrix, $l_{22}$ is scalar, $L_{11}$ is also a nxn matrix. So Cholasky factorization of this B matrix will give us the following: $L_{11}*L_{11}^T=A, L_{11}*L_{21}^T=u, L_{21}*L_{11}^T=u^T$, and $L_{21}*L_{21}^T=1-l_{11}^2$
What to do after this?
Is the factorization done?
If this is done, how can I compute B's complexity(flops) if A is given - lets say it is N?
Thanks.