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 $\mathscr{P}_{3}(\mathbb{D})$, 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.