Weighted inductive means

In this paper we present a unified framework for weighted inductive means on the cone PP of positive definite Hermitian matrices as natural multivariable extensions of two variable weighted means, particularly of metric midpoint operations on PP. It includes some well-known multivariable weighted matrix means: the weighted arithmetic, harmonic, resolvent, Sturm's inductive geometric mean on the Riemannian manifold PP equipped with the trace metric, Log-Euclidean and spectral geometric means. A recursion (or weight additive) formula is derived and applied to find a closed form and basic properties for a weighted inductive mean. An upper bound on the sensitivity, a metric characterization and min and max optimization problems over permutations for the inductive geometric mean are presented. Moreover, we apply the obtained results to a class of midpoint operations of the non-positively curved Hadamard metrics on PP parameterized over Hermitian unitary matrices.