The Riemannian mean of positive matrices. (English) Zbl 1271.15019

Bhatia, Rajendra

Nielsen, Frank (ed.) et al., Matrix information geometry. Selected papers based on the presentations at the Indo-French workshop on matrix information geometries (MIG): Applications in sensor and cognitive systems engineering, Palaiseau, France, February 23–25, 2011. Berlin: Springer (ISBN 978-3-642-30231-2/hbk; 978-3-642-30232-9/ebook). 35-51 (2013).

Summary: The geometric mean of two positive (definite) matrices has been studied for long and found useful in problems of operator theory, quantum mechanics and electrical engineering. For more than two positive matrices, the problem of having an acceptable definition was resolved recently. Among these the object variously called the Riemannian mean, the Karcher mean, the least squares mean, and the barycentre mean is particularly attractive because of intrinsic connections with Riemannian geometry. This mean has also been adopted for applications in image processing, elasticity and other areas. This article is a broad survey of this topic and also reports on some very recent work that is yet to appear.
For the entire collection see [Zbl 1252.94003].

MSC:

15B48	Positive matrices and their generalizations; cones of matrices
26E60	Means
15-02	Research exposition (monographs, survey articles) pertaining to linear algebra

Keywords:

survey article; positive (definite) matrices; Riemannian mean; Karcher mean; least squares mean; barycentre mean

Software:

Matrix Means Toolbox