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Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices |
Canad. J. Math. 56(2004), 776-793
Published:2004-08-01
Printed: Aug 2004
Printed: Aug 2004
- Yongdo Lim
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 matrices 49R50 - Variational methods for eigenvalues of operators (See also 47A75) 15A18 - Eigenvalues, singular values, and eigenvectors 53C3 - unknown classification 53C3 |