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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)


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
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Bas Lemmens
Date Deposited: 14 Jul 2015 08:24 UTC
Last Modified: 09 Dec 2022 09:34 UTC
Resource URI: (The current URI for this page, for reference purposes)

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