  This article focuses on solving the generalized eigenvalue problems (GEP)
arising in the source-free Maxwell equation with magnetoelectric coupling
effects that models three-dimensional complex media. The goal is to compute the
smallest positive eigenvalues, and the main challenge is that the coefficient
matrix in the discrete Maxwell equation is indefinite and degenerate. To
overcome this difficulty, we derive a singular value decomposition (SVD) of the
discrete single-curl operator and then explicitly express the basis of the
invariant subspace corresponding to the nonzero eigenvalues of the GEP.
Consequently, we reduce the GEP to a null space free standard eigenvalue
problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP
and can be solved by the shift-and-invert Arnoldi method without being
disturbed by the null space. Furthermore, the basis of the eigendecomposition
is chosen carefully so that we can apply fast Fourier transformation
(FFT)-based matrix vector multiplication to solve the embedded linear systems
efficiently by an iterative method. For chiral and pseudochiral complex media,
which are of great interest in magnetoelectric applications, the NFSEP can be
further transformed to a null space free generalized eigenvalue problem whose
coefficient matrices are Hermitian and Hermitian positive definite
(HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method
without shifting. Furthermore, the embedded linear system can be solved
efficiently by using the conjugate gradient method without preconditioning and
the FFT-based matrix vector multiplications. Numerical results are presented to
demonstrate the efficiency of the proposed methods.
