I'm having trouble getting the types right for the formula mentioned under the Matrix Function heading. Is f meant to be linear? Is f' meant to be pointwise derivative?

This might help: https://nhigham.com/2020/06/09/what-is-a-matrix-function/

The bidiagonal matrix looks like a chain of binary terms. Is there any relation to monads or possibly lenses, or am I making this up?

In fact it is a common way to represent binary trees for financial derivative valuations (and probably other applications), good eye.

With my limited knowledge, I don't see it. Could you elaborate on how you made that connection?