We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean. © 2018 Elsevier Inc.

Rajendra Bhatia

Department of Mathematics

T. Jain

Y. Lim

Fulltext

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Linear Algebra and Its Applications

Some norm inequalities for matrix means

Norm inequalities related to the matrix geometric mean

An attractive candidate for the geometric mean of m positive definite matrices A 1, ..., A m is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by Lawson and Lim. We give an elementary proof of this property using standard matrix analysis and some counting arguments. We derive some new inequalities for G. One of these says that, for any unitarily invariant norm, {pipe}{pipe}{pipe}G{pipe}{pipe}{pipe} is not bigger than the geometric mean of {pipe}{pipe}{pipe}A 1{pipe}{pipe}{pipe}, ..., {pipe}{pipe}{pipe}A m{pipe}{pipe}{pipe}. © 2011 Springer-Verlag.

Mathematische Annalen

Monotonicity of the matrix geometric mean

The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authors do not readily extend to three matrices and it has been a long-standing problem to define a natural geometric mean of three positive definite matrices. In some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of A and B as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining A and B. This suggests some natural definitions for a geometric mean of three positive definite matrices. We explain the necessary geometric background and explore the properties of some of these candidates. © 2004 Elsevier Inc. All rights reserved.

Journal	Data powered by TypesetLinear Algebra and Its Applications
Publisher	Data powered by TypesetElsevier Inc.
ISSN	00243795