We explain the aims of functional calculus and specialize to polynomials evaluated at endomorphisms. We reconsider the Jordan decomposition and prove it with more generality. Then, we discuss Taylor expansion in the nilpotent part for endomorphisms with separable minimal polynomials, and prove Cayley-Hamilton again for arbitrary fields.
Posts about Cayley-Hamliton.
We want to give a proof of the Cayley-Hamilton Theorem for all commutative rings with unity, which first reduces to the case of the field of complex numbers and then applies a topological argument.