On universal eigenvalues and eigenvectors of hypermatrices
Cheng D, Ji Z (2025)
Publication Type: Journal article
Publication year: 2025
Journal
Book Volume: 362
Article Number: 108126
Journal Issue: 17
DOI: 10.1016/j.jfranklin.2025.108126
Abstract
We introduce the hypervector, obtained via the semi-tensor product (STP) of a finite set of vectors, together with the monic decomposition algorithm (MDA) and the order-reducing map (ORM). Using the hypervector as an eigenvector, we define the universal (U-) eigenvector of a hypermatrix, which extends the concept of the diagonal (D-) eigenvector in the literature and admits a clear geometric interpretation. A unified framework for solving the U-eigenequation is developed. First, the ORM transforms the non-homogeneous U-eigenequation into a homogeneous form. The hypervector representation then converts it into the general matrix eigenequation (GME), which is solved to obtain the eigenvectors as hypervectors. Finally, the MDA decomposes the hypervector into its component vectors, yielding the complete hypermatrix eigenvectors. This approach applies to all known D-eigenequations, substantially reduces computational complexity by converting multi-variable polynomial equations to linear ones, and is validated through numerical experiments in tensor analysis and artificial intelligence, demonstrating clear advantages over existing algorithms.
Authors with CRIS profile
Zhengping Ji
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
China (CN)How to cite
APA:
Cheng, D., & Ji, Z. (2025). On universal eigenvalues and eigenvectors of hypermatrices. Journal of the Franklin Institute-Engineering and Applied Mathematics, 362(17). https://doi.org/10.1016/j.jfranklin.2025.108126
MLA:
Cheng, Daizhan, and Zhengping Ji. "On universal eigenvalues and eigenvectors of hypermatrices." Journal of the Franklin Institute-Engineering and Applied Mathematics 362.17 (2025).
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