Cayley-Hamilton Theorem

Every linear operator satisfies its own characteristic polynomial

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

\chi_T(T)=0,

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

T^n+c_{n-1}T^{n-1}+\cdots+c_0 I=0

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 $\chi_T$ .