Linear Algebra 5: A linear transformation on a finite-dimensional ...

Properties of the characteristic polynomial
• c T (x) is well defined (independent of basis).
• If S = U −1 T U, where U : V → V is <strong>linear</strong> and invertible,
then c S(x) = c T (x).
• Roots of c T (x) are the eigenvalues of T .
• If λ ∈ F is an eigenvalue of T then ∃v ∈ V such that v = 0
and T v = λv, that is, there exists an eigenvector for T with
eigenvalue λ.
• Conversely, if v is an eigenvector for T , so that v = 0 and
∃λ ∈ F such that T v = λv, then c T (λ) = 0.
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