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

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.

Proof sketch: Decompose the $k[x]$-module $V$ into a direct sum of cyclic modules $k[x]/(f_i)$ with $f_1\mid\cdots\mid f_s$. In each cyclic summand, choose the basis $v,xv,\dots,x^{d-1}v$; the action of $x$ is then represented by the companion matrix of the annihilating polynomial. Taking the direct sum basis yields the block diagonal form.