= Minimal and characteristic polynomials from invariant factors
{title2=$m_T=f_r,\qquad c_T=\prod_{j=1}^r f_j$}
Suppose the <rational canonical form> of $T$ is $C(f_1)\oplus\cdots\oplus C(f_r)$, with monic nonconstant <invariant factors of a linear operator> satisfying $f_1\mid\cdots\mid f_r$. Each block has <minimal polynomial> and <characteristic polynomial> $f_j$, because a <cyclic operator has equal minimal and characteristic polynomials>. The <determinant of a block diagonal matrix> is the product of block determinants, giving $c_T=\prod_jf_j$. A <polynomial> kills the direct sum exactly when it kills each block, equivalently when every $f_j$ divides it. Thus $m_T$ is their <least common multiple>, which the divisibility chain reduces to $f_r$. On the zero vector space both polynomials are conventionally one.
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