Rational canonical form theorem

Every linear operator is similar to a block diagonal companion-matrix form determined by invariant factors.

Rational canonical form theorem

Rational canonical form theorem: Let $T$ be a linear map on a finite-dimensional vector space $V$ over a field $k$ . Then there exists a basis of $V$ such that the matrix representation of $T$ is block diagonal with blocks equal to companion matrices $C(f_1),\dots,C(f_s)$ of monic polynomials $f_1,\dots,f_s\in k[x]$ satisfying

f_1 \mid f_2 \mid \cdots \mid f_s.

The polynomials $f_i$ (the invariant factors) are uniquely determined by $T$ . Moreover, $f_s$ is the minimal polynomial of $T$ , and $\prod_i f_i$ is the characteristic polynomial of $T$ .

Conceptually, one views $V$ as a module over the polynomial ring $k[x]$ via $x\cdot v := T(v)$ and applies the structure theorem over a PID (since $k[x]$ is a PID). The companion blocks encode the cyclic summands in the resulting module decomposition.