Krein spaces numerical ranges and their computer generation

Let J be an involutive Hermitian matrix with signature ( t, n - t), 0 <= t <= n, that is, with t positive and n - t negative eigenvalues. The Krein space numerical range of a complex matrix A of size n is the collection of complex numbers of the form xi*JA xi/xi*J xi, with xi epsilon C-n and xi*J xi not equal 0. In this note, a class of tridiagonal matrices with hyperbolical numerical range is investigated. A Matlab program is developed to generate Krein spaces numerical ranges in the finite dimensional case.