Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

We show that for a given set $\mathrm{\Lambda }$ of $nk$ distinct real numbers ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{nk}$ and $k$ graphs on $n$ nodes, $G_0, G_1,\ldots,G_{k-1}$,therearerealsymmetric$n\times n$matrices$A_s$,$s=0,1,\ldots, k$,suchthatthematrixpolynomial$A(z) := A_k z^k + \cdots + A_1 z + A_0$has$\Lambda$asitsspectrum,thegraphof$A_s$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).