Spectral Decomposition and Linearization of Kubo-Ando Means

Abstract

In this paper, we study the structure of Kubo-Ando means on the cone of positive Hermitian matrices over the real numbers, complex numbers, and quaternions. Given a Kubo-Ando mean $\sigma$ with representing function $f$, we obtain an explicit decomposition of $\text{A}\sigma \text{B}$ in terms of the spectrum of ${\text{A}}^{-1}\text{B}$. More precisely, we show that $\text{A}\sigma \text{B}$ can be expressed as a finite linear combination of matrices of the form $\text{A}{\left({\text{A}}^{-1}\text{B}\right)}^k$, with coefficients depending only on $f$ and the eigenvalues of ${\text{A}}^{-1}\text{B}$. We first investigate the linear case and characterize the pairs of matrices for which every Kubo-Ando mean admits an affine representation. We then focus on the cone ${\mathcal{P}}_3(\mathbb{D})$, where we derive explicit formulas for the decomposition coefficients in terms of spectral invariants. Finally, we show that the same techniques extend to a broad class of alternative means, yielding explicit decompositions in the commutative setting and extending recent results of Choi, Kim, and Lim.