On two geometric means and sum of adjoint orbits

TL;DR

This paper explores relationships between different geometric means of matrices, extends these concepts to symmetric spaces and Lie groups, and provides new formulas connecting matrix exponentials and adjoint orbits.

Contribution

It establishes a connection between metric and spectral geometric means via log majorization and extends formulas for matrix exponentials to adjoint orbits in Lie groups.

Findings

Relation between t-metric and t-spectral geometric means via log majorization

Extension of matrix exponential formulas to symmetric spaces

Formulation of adjoint orbit formulas for noncompact semisimple Lie groups

Abstract

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pt\'ak, originally for positive definite matrices. The relation between $t$-metric geometric mean and $t$-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices $X$ and $Y$, So's matrix exponential formula asserts that there are unitary matrices $U$ and $V$ such that $$e^{X/2}e^Ye^{X/2} = e^{UXU^*+VYV^*}.$$ In other words, the Hermitian matrix $\log (e^{X/2}e^Ye^{X/2})$ lies in the sum of the unitary orbits of $X$ and $Y$. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.