Annals of Functional Analysis

Approximation problems in the Riemannian metric on positive definite matrices

Rajendra Bhatia and Tanvi Jain

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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.

Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 118-126.

Dates
First available in Project Euclid: 7 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1396833507

Digital Object Identifier
doi:10.15352/afa/1396833507

Mathematical Reviews number (MathSciNet)
MR3192014

Zentralblatt MATH identifier
1297.15036

Subjects
Primary: 15B48: Positive matrices and their generalizations; cones of matrices
Secondary: 47A58: Operator approximation theory 52A41: Convex functions and convex programs [See also 26B25, 90C25] 53B21: Methods of Riemannian geometry 53B40: Finsler spaces and generalizations (areal metrics)

Keywords
Matrix approximation problem positive definite matrix Riemannian metric convex set Finsler metric

Citation

Bhatia, Rajendra; Jain, Tanvi. Approximation problems in the Riemannian metric on positive definite matrices. Ann. Funct. Anal. 5 (2014), no. 2, 118--126. doi:10.15352/afa/1396833507. https://projecteuclid.org/euclid.afa/1396833507


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