Approximation Problems in the Riemannian Metric on Positive Definite Matrices

There has been considerable work on matrix approximation problems in the space of matrices with Euclidean and unitarily invariant norms. We initiate the study of approximation problems in the space P of all n×n positive definite matrices with the Riemannian metric δ2. Our main theorem reduces the approximation problem in P to an approximation problem in the space of Hermitian matrices and then to that in R. We find best approximants to positive definite matrices from special subsets of P. The corresponding question in Finsler spaces is also addressed. Indian Statistical Institute, New Delhi 110016, India E-mail address: [email protected] E-mail address: [email protected] Date: Received: September 23, 2013; Accepted: December 3, 2013. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 15B48; Secondary 47A58, 52A41, 53B21, 53B40.