Weighted Power Means on Upper Triangular Matrix Groups

We study weighted power means on the Lie group ${\mathrm{UT}}_n$ of real upper triangular matrices with positive diagonal entries. For $t\in \mathbb{R}\setminus \{0\}$, the weighted power mean $P_t(\mathbf{w};A_1,\dots ,A_m)$ is defined by $$X=\sum _{i=1}^m{w}_i(X{\mathrm{\#}}_t{A}_i),X{\mathrm{\#}}_{t}A:=X(X^{-1}A{)}^t\mathrm{.}$$ 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 ${\mathrm{UT}}_2$ and ${\mathrm{UT}}_3$. We prove that $P_t$ converges to the weighted Karcher mean as $t\to 0$. 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.