Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

We show that for a given set of $nk$ distinct real numbers $\Lambda$, 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 proper values $\Lambda$, the graph of $A_s$ is $G_s$ for $s=0,1,\ldots,k-1$, and $A_k$ is an arbitrary nonsingular (positive definite) diagonal matrix. When $k=2$, this solves a physically significant inverse eigenvalue problem for linked vibrating systems.