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 \(L\left( \frac{d}{dt}\right) u=0\), 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 \(Q\in N_{\kappa }^{n \times n}\), that satisfies certain conditions at \(\infty \), and a canonical set of root functions of \(\hat{Q}(z):= -Q(z)^{-1}\), we construct the corresponding Pontryagin space \((\mathcal {K}, [.,.])\), a self-adjoint operator \(A:\mathcal {K}\rightarrow \mathcal {K}\), and an operator \(\Gamma : \mathbb {C}^{n}\rightarrow \mathcal {K}\), 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 \(Q(z):=\hat{L}(z):= -L(z)^{-1}\).