Invariant factors of a linear operator

For a finite-dimensional vector space regarded as an $F[X]$-module through a linear operator, the monic invariant factors $a_1\mid \cdots \mid a_k$ give the decomposition ${⨁}_{i}F[X]/(a_i)$. Their product is the characteristic polynomial and the largest is the minimal polynomial.

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 $c_T={\prod }_j{f}_j$. 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.