Rational canonical form

Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$. Then there are unique monic nonconstant polynomials $f_1, \dots, f_r \in F[t]$ with $f_1 \mid f_2 \mid \cdots \mid f_r$ and the following property. The matrix $A$ is similar to the block diagonal matrix $$\begin{pmatrix} C(f_1) & & \\ & \ddots & \\ & & C(f_r) \end{pmatrix},$$ where $C(f_i)$ is the companion matrix of $f_i$. This block matrix is the rational canonical form of $A$.

Remarks

The polynomials $f_1, \dots, f_r$ are the invariant factors of $A$. The last factor $f_r$ is the minimal polynomial of $A$, and $f_1 f_2 \cdots f_r$ is its characteristic polynomial. Unlike the Jordan form, it needs no field extension: it works over any field. Two matrices over $F$ are similar if and only if they have the same invariant factors.

Explanations

L1 human

The rational canonical form is a standard representative of each similarity class, over any field. It uses companion matrices instead of Jordan blocks. It needs no roots of polynomials, only divisibility of polynomials.