Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 1 / a / Solution

The minimal polynomial $m_T(t)$ is the unique monic polynomial of least positive polynomial degree satisfying $m_T(T)=0$. Such an annihilating polynomial exists without using the Cayley-Hamilton theorem: the vector space $\mathrm{End}(V)$ has dimension of a vector space equal to $n^2$, so $I,T,\dots ,T^{n^2}$ are linearly dependent. Normalize a nonzero annihilating polynomial to be monic and choose one of least degree. Two distinct monic choices of that degree would have an annihilating difference of lower degree, contradicting minimality. A nonzero constant cannot annihilate $T$ because $V\ne 0$.

Choose a basis $B$ and let $A=[T{]}_B$. The characteristic polynomial isIt is independent of the chosen basis. Indeed a change of basis replaces $A$ by a similar matrix $P^{-1}AP$, andby multiplicativity of the determinant. Thus this monic degree-$n$ polynomial is an invariant of the linear transformation itself.