Given a square block matrix $$ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb R^{2d \times 2d}, $$ where $A, B, C, D \in \mathbb R^{d \times d}$.
Is there some kind of a block $QR$ decomposition like $$ M = QR = \begin{bmatrix} Q_1 & Q_2 \\ Q_3 & Q_4 \end{bmatrix} \begin{bmatrix} E & F \\ 0 & G \end{bmatrix} $$ for some $E, F, G \in \mathbb R^{d \times d}$? If yes, can the resulting matrices $E, F, G$ be chosen such that each depends only on a subset of $A, B, C, D$? What is the dependency structure?
Thanks a lot in advance!