Skip to content
DraconiticResearch
Search Sources API
186,086 studies31 sourcesUpdated 10 min ago
PreprintCC BY 4.0arXiv

From Eigenvalues/Eigenvectors of Hypermatrices to Canonical Form of Tensors

  • Daizhan Cheng
  • June Feng
  • Hongsheng Qi
  • Jiandong Zhu
Open at sourcePDFDOICite

Abstract

The rings of non-square matrices based on the dimension-keeping (DK-) semi-tensor product (STP) are considered, where a virtual identity is introduced to make each ring possess an identity element. Using this ring structure, four kinds of eigenvalues/eigenvectors (EEs) of hypermatrices, namely, the ordinary EE (OEE), the universal EE (UEE), the diagonal EE (DEE), and the horizontal diagonal EE (HDEE), are proposed with respect to preassigned matricizings. The Kronecker canonical form (KCF) of non-square pencils is used to calculate the OEEs; the monic decomposition algorithm (MDA) is then applied to extract the UEEs, DEEs, and HDEEs from them. Finally, the KCF of non-square pencils is further used to construct the KCF of a tensor, which reveals all the EEs of the tensor. The KCF of a tensor not only shares the main properties of the Jordan canonical form of matrices but also includes the latter as a special case. Consequently, all the EEs of a hypermatrix are straightforwardly computable via its KCF.

Related

  1. Joint distributions of eigenvectors of symmetric random tensors
    Naoki Sasakura
    arXivPreprintCC BY 4.0Full textMathematical physicsHigh-energy & particle physicsarXiv
  2. Eigenvectors and Eigenvalues of p-Curvature Operators for KZ--type Differential Connections
    Vitaly Tarasov, Alexander Varchenko
    arXivPreprintCC BY 4.0Full textMathematical physicsarXiv
  3. Two-sample tests for principal eigenvalues and eigenvectors in high-dimensional elliptical factor models
    Xinyue Xu, Mengtao Wen, Long Feng
    arXivPreprintCC BY 4.0Full textStatisticsarXiv
  4. Consequences of Matrices Sharing Eigenvalues and Eigenvectors
    Kayla Miller, Steven J. Miller, Fuzhen Zhang
    arXivPreprintCC BY 4.0Full textGeneral mathematicsarXiv
  5. The Universal Nilpotent Cone and a Canonical 1-form
    Marius Kjærsgaard
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{cheng2026from,
  title = {{From Eigenvalues/Eigenvectors of Hypermatrices to Canonical Form of Tensors}},
  author = {Daizhan Cheng and June Feng and Hongsheng Qi and Jiandong Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.00728},
  eprint = {2608.00728},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.00728},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - From Eigenvalues/Eigenvectors of Hypermatrices to Canonical Form of Tensors
AU  - Daizhan Cheng
AU  - June Feng
AU  - Hongsheng Qi
AU  - Jiandong Zhu
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2608.00728
UR  - https://arxiv.org/abs/2608.00728
AB  - The rings of non-square matrices based on the dimension-keeping (DK-) semi-tensor product (STP) are considered, where a virtual identity is introduced to make each ring possess an identity element. Using this ring structure, four kinds of eigenvalues/eigenvectors (EEs) of hypermatrices, namely, the ordinary EE (OEE), the universal EE (UEE), the diagonal EE (DEE), and the horizontal diagonal EE (HDEE), are proposed with respect to preassigned matricizings. The Kronecker canonical form (KCF) of non-square pencils is used to calculate the OEEs; the monic decomposition algorithm (MDA) is then applied to extract the UEEs, DEEs, and HDEEs from them. Finally, the KCF of non-square pencils is further used to construct the KCF of a tensor, which reveals all the EEs of the tensor. The KCF of a tensor not only shares the main properties of the Jordan canonical form of matrices but also includes the latter as a special case. Consequently, all the EEs of a hypermatrix are straightforwardly computable via its KCF.
ER  -