Cayley-Hamilton Theorem

Cayley-Hamilton Theorem: Let $T$ be a linear operator on a finite-dimensional vector space $V$ over a field $F$, and let $\chi_T(t)$ be its characteristic polynomial . Then

meaning that when $\chi_T(t)$ is expanded as a polynomial $\chi_T(t)=t^n+c_{n-1}t^{n-1}+\cdots+c_0$, one has

as operators on $V$ (where $I$ is the identity operator and $0$ is the zero operator).

A key consequence is that the minimal polynomial of $T$ divides $\chi_T$.

Proof sketch: Choose a basis so that $T$ is represented by a matrix $A$. Over the polynomial ring $F[t]$ one has the adjugate identity $(tI-A)\operatorname{adj}(tI-A)=\det(tI-A)\,I$. Interpreting both sides as matrix polynomials and substituting $t=A$ (which makes sense because $A$ commutes with its powers) yields $\chi_A(A)=0$, hence $\chi_T(T)=0$.