Linear Algebra 4$_{th}$ ed. defines the characteristic polynomial for a linear operator as follows:
Let T be a linear operator on a n-dimensional vector space V with order basis, $\beta$. We define the characteristic polynomial, $f(t)$ of T to the characteristic polynomial of $A=[T]_\beta$. That is, $f(t)=|A-tI_n|$
Does $\beta$ need to be a basis composed of eigenvectors? If not, why not? When you need a regular ordered basis of V vs. an ordered basis of eigenvectors.
Thank you!