Annals of Functional Analysis

Approximation problems in the Riemannian metric on positive definite matrices

Rajendra Bhatia and Tanvi Jain

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 7 April 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

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


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.

Export citation


  • T. Ando, Approximation in trace norm by positive semidefinite matrices, Linear Algebra Appl. 71 (1985), 15–21.
  • T. Ando, T. Sekiguchi and T. Suzuki, Approximation by positive operators, Math. Z. 131 (1973), 273–282.
  • R. Bhatia, Matrix Analysis, Springer, 1997.
  • R. Bhatia, Positive Definite Matrices, Princeton University Press, 2007.
  • R. Bhatia, On the exponential metric increasing property, Linear Algebra Appl. 375 (2003), 211-220.
  • N.J. Higham, Matrix nearness problems and applications, in M. J. C. Gover and S. Barnett eds.,Applications of Matrix Theory, Oxford University Press, 1989.
  • F. Nielsen and R. Bhatia, eds., Matrix Information Geometry, Springer, 2013.
  • R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970.