Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices

Lim, Yongdo

Canadian journal of mathematics, Tome 56 (2004) no. 4, pp. 776-793

Résumé

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 $q\,\le \,2$ .

Détail

DOI : 10.4153/CJM-2004-035-5

Mots-clés : 15A48, 49R50, 15A18, 53C3, Matrix approximation, positive definite matrix, geodesic submanifold, Cartan-Hadamard manifold, best approximation, minimal distance function, global tubular neighborhood theorem, Schur complement, metric and spectral geometric mean, Cayley transform

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