Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan–Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. T...

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Veröffentlicht in:Linear algebra and its applications 2018-09, Vol.553, p.129-144
Hauptverfasser: Hiai, Fumio, Lim, Yongdo
Format: Artikel
Sprache:eng
Schlagworte:
(Symplectic) eigenvalue
Cartan barycenter
Center of gravity
Eigenvalues
Linear algebra
Log-majorization
Mathematical analysis
Matrix methods
Positive definite matrix
Probability measure
Random variables
Riemannian trace metric
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Zusammenfassung:In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan–Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2018.04.029