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 ? ? ?, this paper finds that the ratio ? = ?-? ?-? , including infinity if ? = ?, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ?, ? and ? as its three largest eigenvalues. It is shown that such a matrix of size n × n does not exist if n is even and ? is too large or if n is odd and ? is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.