-- Do a 2x2 block matrix multiplication to convince yourself that this preserves the form achieved so far. I don't consider this a proof. Perhaps you have in m

-- Do a 2x2 block matrix multiplication to convince yourself that this preserves the form achieved so far. I don't consider this a proof. Perhaps you have in mind two simple but key properties of reflections about the hyperplane orthogonal to a vector v: (a) The hyperplane of a reflection is the fixed point of the reflection (b) the hyperplane is the orthogonal vector space to the vector space spanned by v. From this two properties it follows that each step of making zeroes does not change previous zeroes. Your claim that for advanced students there is no need to comment about details it is not falsifiable. Citing Mac Lane: A monad is just a monoid in the category of endofunctors. But from a practical point of view one can see the very basic level and simplicity of the definitions and calculations prior to the proof. So at this level of detail I consider that noticing that one must be careful to not destroy previous zeros is matching the level of discourse at the proper level.