  We introduce the concept of mode-k generalized eigenvalues and eigenvectors
of a tensor and prove some properties of such eigenpairs. In particular, we
derive an upper bound for the number of equivalence classes of generalized
tensor eigenpairs using mixed volume. Based on this bound and the structures of
tensor eigenvalue problems, we propose two homotopy continuation type
algorithms to solve tensor eigenproblems. With proper implementation, these
methods can find all equivalence classes of isolated generalized eigenpairs and
some generalized eigenpairs contained in the positive dimensional components
(if there are any). We also introduce an algorithm that combines a heuristic
approach and a Newton homotopy method to extract real generalized eigenpairs
from the found complex generalized eigenpairs. A MATLAB software package TenEig
has been developed to implement these methods. Numerical results are presented
to illustrate the effectiveness and efficiency of TenEig for computing complex
or real generalized eigenpairs.
