Midpoints for Thompson's metric on symmetric cones

Abstract

We characterise the affine span of the midpoints sets, $\mathcal{M}(x,y)$, for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of $\mathcal{M}(x,y)$ in case the associated Euclidean Jordan algebra is simple. In particular, we find for $A$ and $B$ in the cone positive definite Hermitian matrices that \[ \dim({\rm aff}\, \mathcal{M}(A,B))=q^2, \] where $q$ is the number of eigenvalues $\mu$ of $A^{-1}B$, counting multiplicities, such that \[ \mu\neq \max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\}, \] where $\lambda_+(A^{-1}B):=\max \{\lambda\colon \lambda\in\sigma(A^{-1}B)\}$ and $\lambda_-(A^{-1}B):=\min\{\lambda\colon \lambda\in\sigma(A^{-1}B)\}$. These results extend work by Y. Lim [18].