Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

TL;DR

This paper demonstrates that for specified spectral data and graph structures, one can construct structured matrix polynomials with prescribed spectra and graph properties, solving a key inverse spectral problem for linked vibrating systems when k=2.

Contribution

It provides a constructive solution to inverse spectral problems for structured matrix polynomials with prescribed spectra and graph constraints, particularly addressing linked vibrating systems.

Findings

Existence of structured matrix polynomials with given spectra and graph structures.

Solution to a significant inverse eigenvalue problem for linked vibrating systems when k=2.

Construction method applicable to real symmetric matrices with specified graph and spectral properties.

Abstract

We show that for a given set $\Lambda$ of $nk$ distinct real numbers $\lambda_1, \lambda_2, \ldots, \lambda_{nk}$ and $k$ graphs on $n$ nodes, $G_0, G_1,\ldots,G_{k-1}$, there are real symmetric $n\times n$ matrices $A_s$, $s=0,1,\ldots, k$, such that the matrix polynomial $A(z) := A_k z^k + \cdots + A_1 z + A_0$ has $\Lambda$ as its spectrum, the graph of $A_s$ is $G_s$ for $s=0,1,\ldots,k-1$, and $A_k$ is an arbitrary positive definite diagonal matrix. When $k=2$, this solves a physically significant inverse eigenvalue problem for linked vibrating systems (see Corollary 5.3).