On the Inverse Problem of Constructing Symmetric Pentadiagonal Toeplitz Matrices from Three Largest Eigenvalues

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 α = λ−ν μ−ν , 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 one. When such a matrix does exist, a numerical method is proposed for the construction.