= Cyclic operator has equal minimal and characteristic polynomials
{title2=$m_T=c_T$}
On a $d$-dimensional <vector space>, a <cyclic vector> $v$ gives a <basis> $v,Tv,\ldots,T^{d-1}v$. Write $T^dv=-\sum_{j<d}c_jT^jv$ and set $f(t)=t^d+\sum_{j<d}c_jt^j$. Then $f(T)v=0$. Since <polynomials> in $T$ commute with $T$, $f(T)$ kills every basis vector, so $f(T)=0$. No nonzero polynomial of degree less than $d$ can kill $v$, by <linear independence>. Hence the <minimal polynomial> is $f$. The matrix in this basis is $C(f)$, and the <characteristic polynomial of a companion matrix> is $f$ as well.
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