Let be an -dimensional vector space over a field . Prove that similar linear transformations on have the same characteristic and the same minimal polynomial.
Suppose and are similar matrices (i.e. linear transformations) on . Then we have an invertible matrix such that .
Now . So and have the same characteristic polynomial.
Recall that the minimal polynomial of a transformation is the unique monomial generator of in (under the usual action of on ). Thus, to show that and have the same minimal polynomial, it suffices to show that they have the same annihilator in under their respective actions on .
To this end, suppose (where denotes the annihilator induced by ). Then as a linear transformation, we have . Say ; then . But , so that , and thus (left- and right-multiplying by and , respectively) . So as a linear transformation, and we have . The reverse inclusion is similar, so that as desired.