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# Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices

Published:2004-08-01
Printed: Aug 2004
• Yongdo Lim
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## Abstract

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
 Keywords: 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
 MSC Classifications: 15A48 - Positive matrices and their generalizations; cones of matrices49R50 - Variational methods for eigenvalues of operators (See also 47A75)15A18 - Eigenvalues, singular values, and eigenvectors 53C3 - unknown classification 53C3

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