rational canonical form

Overarching theorem

Let $M$ be a finite dimensional vector space over $k$ and $X:M\to M$ a $k$-linear map. There is a decomposition

$$M\simeq M_1\oplus +\cdots +M_K$$

where each $M_i$ is a cyclic $k[x]$-module isomorphic to $k[x]/(q_i)$ for a polynomial $q_i\in k[x]$ such that

$$q_1|\dots |q_K.$$

Moreover, $q_1,\dots ,q_K$ is uniquely determined by $M$ and $q_K$ is the minimal polynomial of $M$.

Proof

Since $M$ is a vector space it is a finitely generated torsion module by this identification. Therefore, by the theorems for finitely generated PID modules, we can use the decomposition into cyclic modules.

Since $q_i|q_K$ for all $i$, we have $q_K(X)=0$. There cannot be a polynomial of degree less than $q_K$ that would annihilate $M$ because it would not annihilate $M_K$. So $q_K$ is the minimal polynomial of $M$.

Rational canonical form

Using this decomposition, $M$ has a decomposition into a rational canonical form which is a basis of $M$ in which $X$ acts by the matrix

$$\left[\begin{array}{ccc}\begin{array}{c}{C}_{q_1}\end{array}& & 0\\ & \ddots & \\ 0& & \begin{array}{c}{C}_{q_K}\end{array}\end{array}\right]$$

Where $C_{q_i}$ is the companion matrix for the $i^{th}$ cyclic component. Moreover, two $k$-linear map $X,Y:M\to M$ are similar if and only if they have the same rational canonical form.

In a bigger field

Let $k\subset \mathbb{K}$ be a subfield and

$$X,Y\in {\text{Mat}}_{n\times n}(k)\subset {\text{Mat}}_{n\times n}(\mathbb{K})$$

1. The rational canonical form of $X$ is the same whether computed in $k$ or $\mathbb{K}$. In particular, the minimal polynomial of $X$ is the same whether computed in $k$ or $\mathbb{K}$

2. The matrices $X$ and $Y$ are similar over $k$ if and only if they are similar over $\mathbb{K}$. In particular, if there exists $P\in {\text{GL}}_n(\mathbb{K})$ such that $X=PY{P}^{-1}$ then there exists $Q\in {\text{GL}}_n(k)$ such that $X=QY{Q}^{-1}$.

Proof

1. The rational canonical form of $X/k$ is defined over $\mathbb{K}$. Since rational canonical form is unique (and determines the minimal polynomial as the largest invariant factor), this prove the first statement.

2. This statement is true by the first statement and that 2 matrices are similar if and only if they have the same rational canonical form.

Computing the rational canonical form

In order to compute the rational canonical form, we generally need to use the characteristic polynomial and the Cayley-Hamilton theorem.