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.