Great answer! A related technique for more general functions is discussed in this answer by Lutz Lehmann and the linked paper.

Related on scicomp.se: scicomp.stackexchange.com/q/42376

You have a clear case of the so-called "impostor syndrome", very common among young academics. For a quick description, see this image and this image, but you can look for more detail on the internet.

The link you report is from the website of Deepmind, the AI's creator. Are you asking for further confirmation? Do you wish to ask if Deepmind is trustworthy as a source?

@LSpice In my mind, the columns of $Q$ are a basis, and the columns of $Z$ are another basis; in this sense this is "a pair of bases". The linear operator $A$ becomes triangular if one considers it with $Z$ as basis for its domain and $Q$ as basis for its codomain, and viceversa for $B$.

@Yc This seems a good answer; please post it as an answer, not as comments though.

Your answer assumes that the matrices are square, but OP is interested also in the rectangular case, as can be seen from the counterexample at the end.

Note that OP's original example can be generalized to $\begin{bmatrix}0 & 0\\ B & symmetric\end{bmatrix}$.

I think this question belongs on History of Science and Mathematics rather than here. The original linked one is a mathematical question, but this one seems a history one to me.