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PreprintCC BY 4.0arXiv

i-conjugate for quaternionic matrices and related properties

  • Cailing Yao
  • Bingzhe Hou
  • Xiaoqi Feng
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Abstract

Motivated by the result that a complex n× n matrix A being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion n× n matrix A is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call i-conjugate. Furthermore, we study the concepts related to i-conjugate and their properties, such as unitary i-congruence, i-conjugate normality and i-Hermicity in Mn(ℍ) as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in Mn(ℂ). Finally, we present a new type of polar decomoposition of quaternion matrices.

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Cite

BibTeXRIS
BibTeX
@misc{yao2026iconjugate,
  title = {{$\boldsymboli$-conjugate for quaternionic matrices and related properties}},
  author = {Cailing Yao and Bingzhe Hou and Xiaoqi Feng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22847},
  eprint = {2609.22847},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22847},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - i-conjugate for quaternionic matrices and related properties
AU  - Cailing Yao
AU  - Bingzhe Hou
AU  - Xiaoqi Feng
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.22847
UR  - https://arxiv.org/abs/2609.22847
AB  - Motivated by the result that a complex n× n matrix A being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion n× n matrix A is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call i-conjugate. Furthermore, we study the concepts related to i-conjugate and their properties, such as unitary i-congruence, i-conjugate normality and i-Hermicity in M_n(ℍ) as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in M_n(ℂ). Finally, we present a new type of polar decomoposition of quaternion matrices.
ER  -