<jats:p>First, we present a method for obtaining a canonical set of root functions and Jordan chains of the invertible matrix polynomial <jats:italic>L</jats:italic>(<jats:italic>z</jats:italic>) through elementary transformations of the matrix <jats:italic>L</jats:italic>(<jats:italic>z</jats:italic>) alone. This method provides a new and simple approach to deriving a general solution of the system of ordinary linear differential equations <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$L\left( \frac{d}{dt}\right) u=0$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mfenced> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>dt</mml:mi> </mml:mrow> </mml:mfrac> </mml:mfenced> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>u</jats:italic>(<jats:italic>t</jats:italic>) is <jats:italic>n</jats:italic>-dimensional unknown function. We illustrate the effectiveness of this method by applying it to solve a high-order linear system of ODEs. Second, given a matrix generalized Nevanlinna function <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Q\in N_{\kappa }^{n \times n}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>κ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, that satisfies certain conditions at <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\infty $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula>, and a canonical set of root functions of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\hat{Q}(z):= -Q(z)^{-1}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Q</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, we construct the corresponding Pontryagin space <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(\mathcal {K}, [.,.])$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mo>[</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>]</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, a self-adjoint operator <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$A:\mathcal {K}\rightarrow \mathcal {K}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, and an operator <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\Gamma : \mathbb {C}^{n}\rightarrow \mathcal {K}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, that represent the function <jats:italic>Q</jats:italic>(<jats:italic>z</jats:italic>) in a Krein–Langer type representation. We illustrate the application of main results with examples involving concrete matrix polynomials <jats:italic>L</jats:italic>(<jats:italic>z</jats:italic>) and their inverses, defined as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Q(z):=\hat{L}(z):= -L(z)^{-1}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>L</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>.</jats:p>

Let S be a symmetric linear relation in the Pontyagin space (K, [., .]) and let Π = (H,Γ0,Γ1) be the corresponding boundary triple. We prove that the corresponding Weyl function Q satisfies Q ∈ Nκ(H). Conversely, for regular Q ∈ Nκ(H), we find linear relation S ( A, where A is representing self-adjoint linear relation of Q, and we prove that Q is the Weyl function of the relation S. We also prove  = kerΓ1, where  is the representing relation of the Q̂ := −Q−1. In addition, if we assume that the derivative at infinity Q ′ (∞) := lim z→∞ zQ(z) is a boundedly invertible operator then we are able to decompose A,  and S in terms of S, i.e. we express relation matrices of A,  and S in terms of S, which is a bounded operator in this case.