---
tag: 042JWAR3Q0
kind: statement
role: theorem
name: Rational canonical form
aliases: [frobenius-normal-form]
msc: [15A21]
uses: [0VH1TC1PWF, 027KDW2K72, 0M0SNKZ482]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_normal_form
    locator: General case and theory
same_as:
  wikidata: Q1469423
  wikipedia: Frobenius normal 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|0E89HQATM8]] to the [[block diagonal matrix|001T5FJDC1]]
$$\begin{pmatrix} C(f_1) & & \\ & \ddots & \\ & & C(f_r) \end{pmatrix},$$
where $C(f_i)$ is the [[companion matrix|0ZCWPNZTQ3]] 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.