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PreprintCC BY-NC-SA 4.0arXiv

The Karcher Mean on 2× 2 Triangular Matrices with Positive Diagonal Entries

  • Huajun Huang
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Abstract

Let UTn denote the group of n× n real upper triangular matrices with positive diagonal entries. We derive a closed-form analytic expression for the Karcher mean on UT2 and obtain a sharp convergence threshold for the projection mean iteration with fixed initial diagonal, together with a sufficient convergence condition for arbitrary initial points. In addition, we obtain explicit formulas for the log-Euclidean mean and establish a Lie--Trotter type formula for the Karcher mean on UT2. For UTn we derive the diagonal and first superdiagonal entries of every solution of the Karcher equation and transfer nonconvergence examples from UT2.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026karcher,
  title = {{The Karcher Mean on $2\times 2$ Triangular Matrices with Positive Diagonal Entries}},
  author = {Huajun Huang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37018},
  eprint = {2609.37018},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37018},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - The Karcher Mean on 2× 2 Triangular Matrices with Positive Diagonal Entries
AU  - Huajun Huang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37018
UR  - https://arxiv.org/abs/2609.37018
AB  - Let UT_n denote the group of n× n real upper triangular matrices with positive diagonal entries. We derive a closed-form analytic expression for the Karcher mean on UT_2 and obtain a sharp convergence threshold for the projection mean iteration with fixed initial diagonal, together with a sufficient convergence condition for arbitrary initial points. In addition, we obtain explicit formulas for the log-Euclidean mean and establish a Lie--Trotter type formula for the Karcher mean on UT_2. For UT_n we derive the diagonal and first superdiagonal entries of every solution of the Karcher equation and transfer nonconvergence examples from UT_2.
ER  -