The Companion Matrix of a Linear Homogeneous nth Order ODE

We will now discuss some of the theory regarding linear homogeneous nth order ODEs. Recall that a linear homogeneous $n^{\mathrm{th}}$ order ODE can be written as:

(1)

\begin{align} \quad y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0 \end{align}

We can rewrite this linear homogeneous $n^{\mathrm{th}}$ order ODE as a system of $n$ first order ODEs. We do this by letting:

(2)

\begin{align} \quad x_1 &= y \\ \quad x_2 &= y' \\ \quad & \vdots \\ \quad x_n &= y^{(n-1)} \end{align}

Then by differentiating each of these equations we get:

(3)

\begin{align} \quad x_1' &= x_2 \\ \quad x_2' &= x_3 \\ \quad & \vdots \\ \quad x_n' &= -a_0(t)x_1 -a_1(t)x_2 - ... - a_{n-1}(t)x_n \end{align}

If $A(t)$ is the coefficient matrix of this system, then we can write this system in the form:

(4)

\begin{align} \quad \mathbf{x}' = A(t) \mathbf{x} \end{align}

The coefficient matrix of this system $A(t)$ described above is significant and we give a special name which we define below.

Definition: The Companion Matrix of an $n^{\mathrm{th}}$ order linear homogeneous equation $y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0$ is the matrix $A(t) = \begin{bmatrix} 0 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 1 \\ -a_0(t) & -a_1(t) & -a_2(t) & -a_3(t) & \cdots & a_{n-1}(t) \end{bmatrix}_{n \times n}$.

For example, consider the following third order linear homogeneous ODE:

(5)

\begin{align} \quad y''' + 2ty'' + 4t^2y' + 3y = 0 \end{align}

Then the companion matrix of third order linear homogeneous ODE is:

(6)

\begin{align} \quad A(t) = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -3 & -4t^2 & -2t \end{bmatrix} \end{align}