Spectrum of the Jacobi tau approximation for the second derivative operator

It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the symmetric case of the Gegenbauer Tau method where the range of parameters included in the theorems can be extended and characteristic polynomials given by successive order approximations interlace. This includes the common Chebyshev and Legendre, Tau and Galerkin methods. The characteristic polynomials for the Gegenbauer Tau method are shown to obey three term recurrences plus a constant term which vanishes for the Legendre Tau and Galerkin cases. These recurrences are equivalent to a tridiagonal plus one row matrix structure. The spectral integration formulation of the Gegenbauer Tau method is shown to lead directly to that fundamental and well-conditioned tridiagonal plus one row matrix structure. A Matlab code is provided.