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 merged and characteristic polynomials given by successive order approximations interlace. This includes the common Chebyshev and Legendre tau and Galerkin methods.