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 Jordan decomposition.
We show some kind of universal property for the Jordan decomposition of an endomorphism of a finite dimensional vector space.