Reconstruction of Structured Quadratic Pencils from Eigenvalues on Ellipses and Parabolas

Department of Mathematics, University of Texas at Brownsville One University Boulevard, Brownsville, TX 78575, USA

^⋆ Corresponding author. E-mail: vesselin.vatchev@utb.edu

Abstract

In the present paper we study the reconstruction of a structured quadratic pencil from eigenvalues distributed on ellipses or parabolas. A quadratic pencil is a square matrix polynomial

QP(λ) = M λ²+Cλ +K,

where M, C, and K are real square matrices. The approach developed in the paper is based on the theory of orthogonal polynomials on the real line. The results can be applied to more general distribution of eigenvalues. The problem with added single eigenvector is also briefly discussed. As an illustration of the reconstruction method, the eigenvalue problem on linearized stability of certain class of stationary exact solution of the Navier-Stokes equations describing atmospheric flows on a spherical surface is reformulated as a simple mass-spring system by means of this method.