  In the first part of the paper, we address an invertible matrix polynomial
$L(z)$ and its inverse $\hat{L}(z) := -L(z)^{-1}$. We present a method for
obtaining a canonical set of root functions and Jordan chains of $L(z)$ through
elementary transformations of the matrix $L(z)$ alone. This method provides a
new and simple approach to deriving a general solution of the system of
ordinary linear differential equations $L\left(\frac{d}{dt}\right)u=0$ using
only elementary transformations of the corresponding matrix polynomial $L(z)$.
  In the second part of the paper, given a matrix generalized Nevanlinna
function $Q\in N_{\kappa }^{n \times n}$ and a canonical set of root functions
of $\hat{Q}(z) := -Q(z)^{-1}$, we provide an algorithm to determine a specific
Pontryagin space $(\mathcal{K}, [.,.])$, a specific self-adjoint operator
$A:\mathcal{K}\rightarrow \mathcal{K}$ and an operator $\Gamma:
\mathbb{C}^{n}\rightarrow \mathcal{K}$ that represent the function $Q$ in a
Krein-Langer type representation. We demonstrate the main results through
examples of linear systems of ODEs.
