Matrix geometric means

Matrix geometric means

(to be updated...)

What is... an inductive mean?, Notices of the American Mathematical Society 70.11 (2023): 1851-1855. (text with two typographic errors corrected): An inductive mean is a mean defined as a limit of a convergence sequence of other means like the arithmetic-geometric mean.
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Arithmetic-harmonic mean (AHM) = *inductive mean*

Positive reals: AHM= geometric mean
Complex numbers: AHM modulus = geometric mean, AHM argument = arithmetic mean
Positive-definite matrices, AHM = Riemannian geometric mean (wrt trace metric)

Many ways to generalize scalar to matrix geometric means:

Log-Euclidean mean (quasi-arithmetic mean) but not operator monotone
Inductive arithmetic-harmonic mean
Solution of a matrix equation yielding power means + taking limits

Matrix means and monotone operators: