Definition of characteristic polynomial of a linear transformation

Definition: Let (T: V o V) be a linear transformation on an (n)-dimensional vector space, and let (A) be the matrix representation of (T) with respect to some basis. The characteristic polynomial of (T) is:
[p_T(\lambda) = \det(A - \lambda I)]

This is a polynomial of degree (n) in (\lambda). The eigenvalues of (T) are exactly the roots of the characteristic polynomial.

Example: For (A = egin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}):
[p(\lambda) = \detegin{pmatrix} 2-\lambda & 1 \ 1 & 2-\lambda \end{pmatrix} = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = (\lambda-1)(\lambda-3)]
The eigenvalues are (\lambda = 1) and (\lambda = 3).