Module IV·Article II·~1 min read
Canonical Forms: Rational and Real
Linear Operators and the Jordan Form
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Canonical Forms
Real Matrices with Complex Eigenvalues
A real matrix may have complex eigenvalues, which occur in pairs: $\lambda = \alpha \pm \beta i$. In the real canonical form, instead of complex cells, there appear real $2 \times 2$ blocks $\begin{bmatrix} \alpha & -\beta \ \beta & \alpha \end{bmatrix}$ (rotation-stretch matrix).
Example: the rotation matrix $\begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix}$ — two complex conjugate eigenvalues $e^{\pm i\theta}$.
Rational Canonical Form
Over an arbitrary field (not necessarily algebraically closed), the Jordan form may not exist. The rational canonical form exists over any field.
It is based on invariant factors — divisors of the elementary divisors.
Companion Matrix
For the polynomial $p(t) = t^n + a_{n-1} t^{n-1} + \ldots + a_1 t + a_0$ the companion matrix is:
$ C(p) = \begin{bmatrix} 0 & 0 & \ldots & 0 & -a_0 \ 1 & 0 & \ldots & 0 & -a_1 \ 0 & 1 & \ldots & 0 & -a_2 \ \vdots & \vdots & & \vdots & \vdots \ 0 & 0 & \ldots & 1 & -a_{n-1} \end{bmatrix}. $
The characteristic and minimal polynomials of $C(p)$ are both equal to $p(t)$. This is the main building block of the rational canonical form.
Application in Control Theory
The Jordan form and the rational canonical form are key tools in control theory. The Jordan form of the system matrix $x' = Ax$ determines the qualitative behavior: stability (all $\operatorname{Re} \lambda < 0$), neutrality ($\operatorname{Re} \lambda = 0$), instability (some $\operatorname{Re} \lambda > 0$).
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