I have a system of differential equations which can be written as $R(\frac{d}{dt})w=0$, where $$R(\xi)=\begin{bmatrix}6-5\xi+\xi^{2} & -3+\xi \\ 2-3\xi+\xi^{2} & -1+\xi\end{bmatrix}$$
I want to find the input/output representation for it.
I understand that $R(\xi)$ is supposed to be of the form $R(\xi)=\begin{bmatrix}-Q(\xi) & P(\xi)\end{bmatrix}$
So I want to find such $P(\xi)\in\mathbb{R}^{p\times p}[\xi]$ and $Q(\xi)\in\mathbb{R}^{p\times m}[\xi]$ which satisfy:
$\det P(\xi)\ne 0$ and $P^{-1}(\xi)Q(\xi)$ is a matrix of proper rational functions.
I can partition $w$ as $$w=\begin{bmatrix}u \\ y\end{bmatrix}$$
So that $R(\frac{d}{dt})w=0$ can be written as $P(\frac{d}{dt})y=Q(\frac{d}{dt})u$, which is the input/output representation...I'm having trouble with the first part though, attaining those two matrices.
Note that
$$ R(\xi)=\begin{bmatrix}(\xi - 2)(\xi - 3) & \xi - 3 \\ (\xi - 2)(\xi - 1) & \xi - 1 \end{bmatrix}$$
and $\det R(\xi) = 0$. So it is actually one set of equation. Assuming the correct partition of $w$ is $w = [y ~~ u]^T$ (otherwise it would be non-proper) input/output representation becomes
$$ \frac{y}{u} = \frac{1}{2 - \xi} $$
Edit: Let us write the equations separately:
$$ \begin{align} (\xi - 2)(\xi - 3) y + (\xi - 3) u &= 0 \\ (\xi - 2)(\xi - 1) y + (\xi - 1) u &= 0 \end{align} $$
We need to find $y/u$ that satisfies both of the above equations. But these equations are linearly dependant, so the result follows.