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A real persymmetric Jacobi matrix of order $n$ whose eigenvalues are $2k^2$$(k=0, ..., n-1)$ is presented, with entries given as explicit functions of$n$. Besides the possible use for testing forward and inverse numericalalgorithms, such a matrix is especially relevant for its connection with thedynamics of a mass-spring chain, which is a multi-purpose prototype model.Indeed, the mode frequencies being the square roots of the eigenvalues of theinteraction matrix, one can shape the chain in such a way that its dynamics beperfectly periodic and dispersionless.

Persymmetric Jacobi matrices with square-integer eigenvalues and dispersionless mass-spring chains