On the inverse problem of constructing symmetric pentadiagonal Toeplitz matrices from their three largest eigenvalues

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The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem—can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? Given three real numbers ν ⩽ µ ⩽ λ, this paper finds that the ratio , including infinity if µ = ν, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ν, µ and λ as its three largest eigenvalues. It is shown that such a matrix of size n × n does not exist if n is even and α is too large or if n is odd and α is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.