Osaka Journal of Mathematics

Midpoints for Thompson's metric on symmetric cones

Bas Lemmens and Mark Roelands

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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].

Article information

Osaka J. Math., Volume 54, Number 1 (2017), 197-208.

First available in Project Euclid: 3 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C22: Geodesics [See also 58E10]
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices


Lemmens, Bas; Roelands, Mark. Midpoints for Thompson's metric on symmetric cones. Osaka J. Math. 54 (2017), no. 1, 197--208.

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