This last formula is referred to as the generalised Pythagorean theorem. How is it derived?

This only applies to square matrices, but here's how I think of it. If you have a finite square matrix M representing a transformation V -> V (we don't lose any generality here, since a square matrix V -> W will automatically give you two lists of bases and a way to correspond them, essentially telling you what the right isomorphism is), you can think of the minor as "if I 'delete' a vector b from my basis, and look at the span of the remaining vectors to get a new space W, what happens to M?" If pi projects from V into W, then we can phrase it as looking for a matrix for the composite pi after M. I'm not sure of any nice connections with the matrix representations otherwise, but this can obviously be generalized to any projection pi.

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