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 numerical algorithms, such a matrix is especially relevant for its connection with the dynamics of a mass-spring chain, which is a multi-purpose prototype model. Indeed, the mode frequencies being the square roots of the eigenvalues of the interaction matrix, one can shape the chain in such a way that its dynamics be perfectly periodic and dispersionless.
Persymmetric Jacobi matrices with square-integer eigenvalues and dispersionless mass-spring chains
Ruggero Vaia;
2020
Abstract
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 numerical algorithms, such a matrix is especially relevant for its connection with the dynamics of a mass-spring chain, which is a multi-purpose prototype model. Indeed, the mode frequencies being the square roots of the eigenvalues of the interaction matrix, one can shape the chain in such a way that its dynamics be perfectly periodic and dispersionless.File in questo prodotto:
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Descrizione: Persymmetric Jacobi matrices with square-integer eigenvalues and dispersionless mass-spring chains
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