Approximation problems in the Riemannian metric on positive definite matrices

Abstract

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 $\mathbb{P}$ of all $n\times n$ positive definite matrices with the Riemannian metric $\delta_2$. Our main theorem reduces the approximation problem in $\mathbb{P}$ to an approximation problem in the space of Hermitian matrices and then to that in $\mathbb{R}^n$. We find best approximants to positive definite matrices from special subsets of $\mathbb{P}$. The corresponding question in Finsler spaces is also addressed.