**Theorem (Cayley-Hamilton):** Every linear transformation \(T: V 	o V\) on a finite-dimensional vector space satisfies its own characteristic equation. That is, if \(p_T(\lambda)\) is the characteristic polynomial of \(T\), then:
\[p_T(T) = 0\]

where the polynomial is evaluated at \(T\) (replacing \(\lambda\) with \(T\) and the constant term with that scalar times the identity transformation).

**Intuition:** The characteristic polynomial, when applied to the transformation itself, annihilates it. This means the minimal polynomial always divides the characteristic polynomial.

**Example:** For \(A = egin{pmatrix} 1 & 2 \ 3 & 0 \end{pmatrix}\), the characteristic polynomial is \(p(\lambda) = \lambda^2 - \lambda - 6\). The Cayley-Hamilton theorem says \(A^2 - A - 6I = 0\), which can be verified directly.