I'm looking at QR decomposition: any complex square matrix $B$ can be rewritten as $$B = QR \rightarrow R = Q^\dagger B$$
where $Q$ is unitary and $R$ is an upper triangular matrix. Similarly, any complex square matrix $A$ can be rewritten as $$A = QL \rightarrow L = Q^\dagger A$$ where $L$ is a lower triangular matrix
If I have two square matrices $A, B$ such that $A^\dagger B$ is an upper triangular matrix ($\dagger$ is conjugate transpose), can I conclude that there is a unitary $Q$ such that $Q^\dagger A$ is lower triangular and $Q^\dagger B$ is upper triangular?
The converse seems true: if $Q^\dagger A$ is lower triangular and $Q^\dagger B$ is upper triangular then $$A^\dagger B = A^\dagger Q Q^\dagger B$$ which is an upper triangular matrix by construction.