Cayley–Hamilton theorem

A square matrix satisfies its own characteristic polynomial.

Cayley–Hamilton theorem

Cayley–Hamilton theorem: Let $T:V\to V$ be a linear operator on a finite-dimensional vector space $V$ . Let $p_T(t)$ be the characteristic polynomial of $T$ . Then

p_T(T)=0,

meaning that substituting $T$ into its own characteristic polynomial yields the zero operator on $V$ .

Equivalently, if $A$ is an $n\times n$ matrix and $p_A(t)=\det(tI-A)$ (defined using the determinant ), then $p_A(A)=0$ . One consequence is that the minimal polynomial divides the characteristic polynomial, linking algebraic identities of $T$ to its eigenvalues .