  An efficient Jacobi-Galerkin spectral method for calculating eigenvalues of
Riesz fractional partial differential equations with homogeneous Dirichlet
boundary values is proposed in this paper. In order to retain the symmetry and
positive definiteness of the discrete linear system, we introduce some properly
defined Sobolev spaces and approximate the eigenvalue problem in a standard
Galerkin weak formulation instead of the Petrov-Galerkin one as in literature.
Poincar\'{e} and inverse inequalities are proved for the proposed Galerkin
formulation which finally help us establishing a sharp estimate on the
algebraic system's condition number. Rigorous error estimates of the
eigenvalues and eigenvectors are then readily obtained by using Babu\v{s}ka and
Osborn's approximation theory on self-adjoint and positive-definite eigenvalue
problems. Numerical results are presented to demonstrate the accuracy and
efficiency, and to validate the asymptotically exponential oder of convergence.
Moreover, the Weyl-type asymptotic law $ \lambda_n=\mathcal{O}(n^{2\alpha})$
for the $n$-th eigenvalue $\lambda_n$ of the Riesz fractional differential
operator of order $2\alpha$, and the condition number $N^{4\alpha}$ of its
algebraic system with respect to the polynomial degree $N$ are observed.
