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Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices

Published online by Cambridge University Press:  20 November 2018

Yongdo Lim*
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea e-mail: ylim@knu.ac.kr
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Abstract

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We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold $\text{Sym(}p\text{,}\,\mathbb{R}{{\text{)}}^{++}}\,\times \,\text{Sym(}q\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ block diagonally embedded in $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\,\le \,2$ or $q\,\le \,2$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2004

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