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PreprintCC BY 4.0arXiv

Unitary similarity for dual complex matrices and dual quaternion matrices

  • Cailing Yao
  • Bingzhe Hou
  • Xiaoqi Feng
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Abstract

We study unitary similarity of dual complex matrices and dual quaternion matrices. First, we characterize unitary similarity in terms of the existence of a skew-Hermitian solution to a Sylvester-type matrix equation. For 2×2 dual complex matrices, we obtain a complete three-trace criterion when the standard part is nonnormal and analyze the limitations of trace-word criteria when the standard part is normal. We then prove, for every n≥1, that two n× n dual quaternion matrices are unitarily similar if and only if their dual complex adjoint matrices are unitarily similar, and we extend this result to finite families. As a scalar consequence, we derive a finite Trace-word criterion when the standard part is nonreal and exhibit its failure when the standard part is real.

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Cite

BibTeXRIS
BibTeX
@misc{yao2026unitary,
  title = {{Unitary similarity for dual complex matrices and dual quaternion matrices}},
  author = {Cailing Yao and Bingzhe Hou and Xiaoqi Feng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30060},
  eprint = {2610.30060},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30060},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Unitary similarity for dual complex matrices and dual quaternion matrices
AU  - Cailing Yao
AU  - Bingzhe Hou
AU  - Xiaoqi Feng
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30060
UR  - https://arxiv.org/abs/2610.30060
AB  - We study unitary similarity of dual complex matrices and dual quaternion matrices. First, we characterize unitary similarity in terms of the existence of a skew-Hermitian solution to a Sylvester-type matrix equation. For 2×2 dual complex matrices, we obtain a complete three-trace criterion when the standard part is nonnormal and analyze the limitations of trace-word criteria when the standard part is normal. We then prove, for every n≥1, that two n× n dual quaternion matrices are unitarily similar if and only if their dual complex adjoint matrices are unitarily similar, and we extend this result to finite families. As a scalar consequence, we derive a finite Trace-word criterion when the standard part is nonreal and exhibit its failure when the standard part is real.
ER  -