Tensor eigenvalue problems have found important applications in automatic control, statistical data analysis, diffusion tensor imaging, image authenticity verification, spectral hypergraph theory and quantum entanglement, etc. The concept of mode-$k$ generalized eigenvalues and eigenvectors of a tensor is introduced and some properties of such eigenpairs are proved. In particular, an upper bound for the number of equivalence classes of generalized tensor eigenpairs using mixed volume is derived. Based on this bound and the structures of tensor eigenvalue problems, two homotopy continuation type algorithms to solve tensor eigenproblems are proposed. 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). An algorithm that combines a straightforward approach and a Newton homotopy method is introduced to extract real generalized eigenpairs from the available complex generalized eigenpairs. A MATLAB software package \texttt{TenEig 1.1} has been developed to implement these methods. Numerical results are presented to illustrate the effectiveness and efficiency of \texttt{TenEig 1.1} for computing complex or real generalized eigenpairs.
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