Power Mean, Karcher Mean, and Log-Euclidean Mean on Positive Constant Upper Triangular Matrices

Abstract

We study weighted power, Karcher, and log-Euclidean means on the Lie group CP_n of real upper triangular matrices with a constant positive diagonal. Building on the general triangular-group theory, we use the central decomposition CP_n \cong \mathbb R_+ -by-UP_n to separate the scalar power mean from a unipotent mean with reweighted coefficients. We prove that the power mean extends real-analytically through order zero, where it equals the Karcher mean. We establish exterior-power covariance for the Karcher and log-Euclidean means, and tensor-product identities for all three means, distinguishing product weights from paired inputs. Finite nilpotent functional calculus yields concise formulas on CP_2 and CP_3, where the Karcher and log-Euclidean means coincide. In dimension four, their difference is exactly a central cubic commutator, giving a necessary and sufficient equality criterion and an explicit counterexample. In every dimension, the same commutator is the leading term of their small-dispersion difference.