Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

Abstract

We show that for a given set $\Lambda$ of $nk$ distinct real numbers ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{nk}$ and $k$ graphs on $n$ nodes, $G_0,G_1,\dots ,G_{k-1}$, there are real symmetric $n\times n$ matrices $A_s$, $s=0,1,\dots ,k$, 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 $s=0,1,\dots ,k-1$, 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).

21 pages, 26 references