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PreprintCC BY 4.0arXiv

The ϕ-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization

  • Cailing Yao
  • Bingzhe Hou
  • Yue Xin
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Abstract

Let ϕ be a quaternion of modulus $1$. In this article, we study some topics related to ϕ-conjugation for quaternionic matrices, including ϕ-Hermitian matrices, ϕ-conjugate normal matrices, unitary ϕ-congruence and ϕ-HSH decomposition (decomposition of a ϕ-Hermitian matrix and a skew ϕ-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion ϕ-Hermitian matrices for all unit quaternion ϕ. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.

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Cite

BibTeXRIS
BibTeX
@misc{yao2026ϕconjugation,
  title = {{The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization}},
  author = {Cailing Yao and Bingzhe Hou and Yue Xin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.22825},
  eprint = {2609.22825},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22825},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The ϕ-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization
AU  - Cailing Yao
AU  - Bingzhe Hou
AU  - Yue Xin
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.22825
UR  - https://arxiv.org/abs/2609.22825
AB  - Let ϕ be a quaternion of modulus $1$. In this article, we study some topics related to ϕ-conjugation for quaternionic matrices, including ϕ-Hermitian matrices, ϕ-conjugate normal matrices, unitary ϕ-congruence and ϕ-HSH decomposition (decomposition of a ϕ-Hermitian matrix and a skew ϕ-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion ϕ-Hermitian matrices for all unit quaternion ϕ. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.
ER  -