Cycle/Minimal polynomial/Exercise

Let the cycle ${}1\mapsto 2\mapsto 3\mapsto \ldots \mapsto n\mapsto 1$ be given, and let ${}M$ denote the corresponding ${}n\times n$-permutation matrix over a field ${}K$.

a) Let ${}P\in K[X]$ be a polynomial of degree ${}<n$. Establish a formula for ${}(P(M))(e_{1})$.

b) Determine the minimal polynomial of ${}M$.

c) Give an example for an endomorphism ${}\varphi$ on a real vector space ${}V$ with different vectors ${}v_{1},v_{2},v_{3}\in V$ such that ${}\varphi (v_{1})=v_{2}$, ${}\varphi (v_{2})=v_{3}$ and ${}\varphi (v_{3})=v_{1}$ holds, and such that the minimal polynomial of ${}\varphi$ is not ${}X^{3}-1$.

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