Rational Canonical Form

A unique form used to determine if two square matrixes are similar.

Definition

1. With $T\in \mathcal{L}(V)$
2. Then. $[T{]}_{\beta }$ is the rational canonical form if:

[T] {\beta} = \left[\begin{array}{cc} C(p{1}) & 0 & \dots & 0\ 0 & C(p_{2}) & \ddots & \vdots\ \vdots & \ddots & \ddots & 0\ 0 & \dots & 0 & C(p_{j}) \end{array}\right]

Where $C(p_{i})$ are [[Matrix Definition of Cyclic Subspace|Companion Matrix]] of polynomial $p_{i}$ # Guides - [[Finding Rational Form of Matrix]] - [[Finding Rational Form of Matrix Example 2]] # Theorems - [[Matrixes are Similar if Rational Canonical Form is Same]]