The Cayley-Hamilton Theorem and the Jordan Decomposition

(1) mT (x) = p1 (x) s1 · · · pk (x)k These irreducible factors can then be used to construct certain polynomials f1 (x) , . . . , fk (x) and corresponding operators Ei ≡ fi (T ) which can be used to decompose the vector space V into a direct sum of T -invariant subspaces V = V1 ⊕ V2 ⊕ · · · ⊕ Vk Morever, we have both Vi = Ei (V ) (the image of V under Ei) and Vi = ker (pi (T ) i) (the kernel of the operator pi (T ) i). Since each Vi is T -invariant v ∈ Vi ⇒ T (v) ∈ Vi it follows that if we construct a basis by V by first choosing bases for each subspace Vi and then adjoining these bases to get a basis B for the entire vector space V , then with respect to B, the matrix T will take the block diagonal form