Weighted Power Means on Upper Triangular Matrix Groups
Abstract
We study weighted power means on the Lie group UTn of real upper triangular matrices with positive diagonal entries. For t∈ℝ∖{0}, the weighted power mean Pt(w; A1,…,Am) is defined by X=∑i=1m wi(X#tAi), X#tA:=X(X−1A)t. Using divided differences, analytic matrix functions, and the Parlett recurrence, we prove existence, uniqueness, and real-analytic dependence of Pt on the data and the weights. We derive finite entrywise and block-recursive algorithms, establish structural properties of the mean, and obtain explicit formulas for the cases UT2 and UT3. We prove that Pt converges to the weighted Karcher mean as t→ 0. We also establish a Lie--Trotter formula and derive second-order asymptotic expansions for the power, Karcher, and log-Euclidean means, showing that the parameter t first enters the local behavior at second order.
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BibTeX
@misc{huang2026weighted,
title = {{Weighted Power Means on Upper Triangular Matrix Groups}},
author = {Huajun Huang and Jimmie Lawson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24821},
eprint = {2610.24821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24821},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Weighted Power Means on Upper Triangular Matrix Groups
AU - Huajun Huang
AU - Jimmie Lawson
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.24821
UR - https://arxiv.org/abs/2610.24821
AB - We study weighted power means on the Lie group UT_n of real upper triangular matrices with positive diagonal entries. For t∈ℝ∖{0}, the weighted power mean P_t(w; A_1,…,A_m) is defined by X=∑_i=1^m w_i(X#_tA_i), X#_tA:=X(X^−1A)^t. Using divided differences, analytic matrix functions, and the Parlett recurrence, we prove existence, uniqueness, and real-analytic dependence of P_t on the data and the weights. We derive finite entrywise and block-recursive algorithms, establish structural properties of the mean, and obtain explicit formulas for the cases UT_2 and UT_3. We prove that P_t converges to the weighted Karcher mean as t→ 0. We also establish a Lie--Trotter formula and derive second-order asymptotic expansions for the power, Karcher, and log-Euclidean means, showing that the parameter t first enters the local behavior at second order.
ER -