Applications of companion matrices

Question

I'm looking for interesting applications of companion matrices. I can also use the Frobenius Normal Form.

I already covered the Cayley-Hamilton Theorem and the application to linearly recursive sequences and high-order scalar linear differential equations.

Answer 1

Companion matrices can also be used to find the roots of a polynomial equation: the eigenvalues of the Companion Matrix are the roots of the polynomial equation. It is a very robust way of finding the roots of an equation, but not the most efficient (computationally).

Answer 2

Companion forms of matrices are widely used in control theory, for example, in the observable canonical form as well as the controllable canonical form.

For a little more context of these canonical forms, you might read a little on state space representations in control theory and their relationship to transfer functions in control theory.

Answer 3

Companion matrices can be used to give matricial representations of some fields:

Let $k$ be a field and $k(\alpha)$ be a simple algebraic extension. If $A$ is the companion matrix of the minimal polynomial of $\alpha$, then $k(\alpha) \simeq k(A)$.

For example, $\mathbb{C} \simeq \left\{ \left( \begin{matrix} a & -b \\ b & a \end{matrix} \right) \mid a,b \in \mathbb{R} \right\}$.