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.