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

• Published in: arXiv.org 2024-07
• Main Author: Borogovac, Muhamed
• Format: Article
• Language: English
• Subjects: Algorithms; Differential equations; Linear systems; Operators (mathematics); Polynomials; Transformations (mathematics)
• ISSN: 2331-8422

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.