Approximation problems in the Riemannian matric on positive definite metrices. (English) Zbl 1297.15036

Bhatia, Rajendra; Jain, Tanvi

Ann. Funct. Anal. 5, No. 2, 118-126 (2014).

Summary: 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.

MSC:

15B48	Positive matrices and their generalizations; cones of matrices
41A50	Best approximation, Chebyshev systems
47A58	Linear operator approximation theory
53B21	Methods of local Riemannian geometry
53B40	Local differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

matrix approximation problem; positive definite matrix; Riemannian metric; convex set; Finsler metric