Definition of rational form

**Definition:** The **rational canonical form** (or Frobenius normal form) of a square matrix $A$ is a block diagonal matrix where each block is the **companion matrix** of an invariant factor of $A$:

$$R = \begin{pmatrix} C_1 & 0 & \cdots & 0 \\ 0 & C_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & C_k \end{pmatrix}$$

The companion matrix of $p(x) = x^n + c_{n-1}x^{n-1} + \cdots + c_1 x + c_0$ is:

$$C = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{pmatrix}$$

**Advantage over Jordan form:** The rational canonical form exists over any field (not requiring eigenvalues to be in the field). It is unique up to block ordering.Copy