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# Definition of characteristic polynomial of a linear transformation

Put content here**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:
\[p_T(\lambda) = \det(A - \lambda I)\]
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This is a polynomial of degree \(n\) in \(\lambda\). The eigenvalues of \(T\) are exactly the roots of the characteristic polynomial.
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**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 \(\lambda = 3\).

# Parents

* Eigenvalues and eigenvectors