Let us assume we have two matrices $A$, $B$ for which one can run the Cholesky decomposition and we get $U \times U^{T} = A$ and $V \times V^{T} = B$ where $U$ and $V$ are lower triangular matrices.
Knowing that the matrix $\begin{bmatrix} A&B\\ B^T&A \end{bmatrix}$ is positive definite can we compute the Cholesky decomposition of it using only the matrices $U$ and $V$?
Thanks, Bogdan.