On Universal Eigenvalues and Eigenvectors of Hypermatrices

Abstract

A generalized eigenvector of a hypermatrix, called the universal (U-) eigenvector, is proposed, which extended the notion of diagonal (D-) eigenvectors in the literature. Using the semi-tensor product, the homogeneous U-eigenequation can be converted into a general eigenequation of matrix $(A-\lambda B)x=0$. A general technique for solving this equation is proposed, which leads to two kinds of eigenvalues: essential and quasi eigenvalues. The technique to convert nonhomogeneous eigenequation to homogeneous ones is also revealed. Then a hypervector decomposing method, called the monic decomposition algorithm (MDA), is developed. Using the MDA, the U-eigenproblem (including the D-eigenproblem) can be converted into general matrix eigenproblems. Some examples are presented, demonstrating the geometric meaning and potential applications of the U-eigenvalue/eigenvector.