The Cayley-Hamilton theorem for a linear transformation

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.