An approach to root functions of matrix polynomials with applications in differential equations and meromorphic matrix functions

First, we present a method for obtaining a canonical set of root functions and Jordan chains of the invertible matrix polynomial 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 Lddtu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\left( \frac{d}{dt}\right) u=0$$\end{document}, where u(t) is n-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 Q∈Nκn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\in N_{\kappa }^{n \times n}$$\end{document}, that satisfies certain conditions at ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}, and a canonical set of root functions of Q^(z):=-Q(z)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{Q}(z):= -Q(z)^{-1}$$\end{document}, we construct the corresponding Pontryagin space (K,[.,.])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {K}, [.,.])$$\end{document}, a self-adjoint operator A:K→K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:\mathcal {K}\rightarrow \mathcal {K}$$\end{document}, and an operator Γ:Cn→K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma : \mathbb {C}^{n}\rightarrow \mathcal {K}$$\end{document}, that represent the function Q(z) in a Krein–Langer type representation. We illustrate the application of main results with examples involving concrete matrix polynomials L(z) and their inverses, defined as Q(z):=L^(z):=-L(z)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(z):=\hat{L}(z):= -L(z)^{-1}$$\end{document}.