# Midpoints for Thompson's metric on symmetric cones

Lemmens, Bas, Roelands, Mark (2016) Midpoints for Thompson's metric on symmetric cones. Osaka Journal of Mathematics, 54 (1). pp. 197-208. ISSN 0030-6126. (KAR id:49518)

## Abstract

We characterise the affine span of the midpoints sets, $$M(x,y)$$, 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 $$M(x,y)$$ 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 $$dim(aff M(A,B)) = q^2$$, where $$q$$ is the number of eigenvalues $$\mu$$ of $$A^{-1}B$$, counting multiplicities, such that $$\mu ≠ max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\},$$ where $$\lambda_+(A^{-1}B) := max\{\lambda:\lambda \in \sigma(A^{-1}B)\}$$ and $$\lambda_-(A^{-1}B) := min\{\lambda:\lambda \in \sigma(A^{-1}B)\}$$. These results extend work by Y. Lim [18].

Item Type: Article Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics Bas Lemmens 14 Jul 2015 08:24 UTC 19 Feb 2020 12:13 UTC https://kar.kent.ac.uk/id/eprint/49518 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-6713-7683 https://orcid.org/0000-0002-8885-9156