## Annals of Functional Analysis

### 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.

#### Article information

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

Dates
First available in Project Euclid: 7 April 2014

https://projecteuclid.org/euclid.afa/1396833507

Digital Object Identifier
doi:10.15352/afa/1396833507

Mathematical Reviews number (MathSciNet)
MR3192014

Zentralblatt MATH identifier
1297.15036

#### 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

#### References

• 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.