Novel approach to root functions of matrix polynomials with applications in differential equations and meromorphic matrix functions
Muhamed Borogovac
TL;DR
The paper addresses how to extract root functions and Jordan chains for matrix polynomials and how to represent generalized Nevanlinna functions in Krein-Langer form. It introduces an elementary-transformation approach that yields canonical root functions for an invertible matrix polynomial $L(z)$ and its inverse $\hat{L}(z)=-L(z)^{-1}$ via a diagonalization $D(z)=S(z)L(z)T(z)$ with $D(z)=\operatorname{diag}(d_1(z),\dots,d_n(z))$, linking root functions to zeros of $d_i(z)$. A complementary part develops a constructive Krein-Langer realization for $Q\in N_{\kappa}^{n\times n}$, producing a Pontryagin space $(\mathcal{K},[.,.])$, a self-adjoint operator $A$, and a map $\Gamma$ such that $Q(z)=S+\Gamma^{+}(A-z)^{-1}\Gamma$, enabling concrete operator representations and applications to linear ODEs. Together, these methods allow practical computation of eigenstructures and representations without solving high-degree determinant equations, and reveal connections between root functions and pole-cancellation structures in inverse matrix polynomials.
Abstract
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 $Q\in N_{κ}^{n \times n}$ 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 $Γ: \mathbb{C}^{n}\rightarrow \mathcal{K}$ that represent the function $Q$ in a Krein-Langer type representation. We demonstrate the main results through examples of linear systems of ODEs.