The Karcher Mean on Lie Groups of Upper Triangular Matrices

@misc{huang2026karcher,
  title = {{The Karcher Mean on Lie Groups of Upper Triangular Matrices}},
  author = {Huajun Huang and Jimmie Lawson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34267},
  eprint = {2609.34267},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34267},
  note = {CC BY-NC-SA 4.0},
}

TY  - UNPB
TI  - The Karcher Mean on Lie Groups of Upper Triangular Matrices
AU  - Huajun Huang
AU  - Jimmie Lawson
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34267
UR  - https://arxiv.org/abs/2609.34267
AB  - We study the weighted Karcher mean on the Lie group UT_n of upper triangular matrices with positive diagonal entries. Using a divided-difference description of analytic functions on upper triangular matrices and the Parlett recurrence, we prove the existence, uniqueness, and analyticity of the weighted Karcher mean and derive recursive formulas for its computation. The theory yields explicit expressions in low dimensions and formulas for weighted geometric means on UT_n. More generally, the results reveal a close relationship between divided differences, analytic matrix functions, and nonlinear matrix means on triangular Lie groups.
ER  -