rational canonical form

《Algebraic and Analytic Geometry》 · 第 526 页 · Section 7.4「The Rational Canonical Form」，定义段

For every linear operator $T$ on a finite-dimensional vector space $V$, there exists an ordered basis $\beta$ for $V$ such that $[T]_\beta=C_1\oplus\cdots\oplus C_r$, where each $C_i$ is the companion matrix of a polynomial $(\phi(t))^m$ such that $\phi(t)$ is a monic irreducible divisor of the characteristic polynomial of $T$ and $m$ is a positive integer. A matrix representation of this kind is called a rational canonical form of $T$.