2023/10/22 by Daizhan Cheng, Cheng, Daizhan, Zhengping Ji +1
Computer Science · Mathematics · #Matrix Theory and Algorithms #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2310.14266
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-λB)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.