If I have a symmetric, positive definite block matrix there exists the following LU decomposition:
$$\left[\matrix{A && B^\intercal \\ B && C}\right]=\left[\matrix{A^{\frac{1}{2}} && 0 \\ BA^{-\frac{1}{2}} && Q^{\frac{1}{2}}}\right]\left[\matrix{A^{\frac{1}{2}} && A^{-\frac{1}{2}}(B^{\intercal})^{\frac{1}{2}} \\ 0 && Q^{\frac{1}{2}}}\right]$$
where $Q = C - BA^{-1}B^{\intercal}$. Is it possible to derive a similar expression if I have a matrix subdivided into more than two blocks per row or column, for example:
$$\left[\matrix{A && B^\intercal && D^\intercal \\ B && C && E^\intercal \\ D && E && F}\right]$$
If yes, what would a decomposition equivalent to the one above look like?