Spectra of tridiagonal Toeplitz matrices

This interactive picture is inspired by joint projects developed by
J. M. Bogoya Ramírez, A. Böttcher, S. Grudsky, and E. Maximenko.
The mathematical result is well known.

We consider tridiagonal Toeplitz matrices
generated by Laurent polynomials of the form

      a(t) = a₋₁t⁻¹+a₀t⁰+a₁t¹.

Choose the coefficients:

The black ellipse (or segment) is the image of the symbol.

The red segment is the Schmidt and Spitzer set.

The yellow points are the eigenvalues of the Toeplitz matrix T_n(a).

Select the order of the matrix: