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On the monotonicity of weighted power means for matrices

Authors: 

Trung Hoa Dinh, Raluca Dumitru, Jose A. Franco

Source title: 
Linear Algebra and its Applications, 527: 128-140, 2017 (ISI)
Academic year of acceptance: 
2016-2017
Abstract: 

In this article, we provide an alternate proof of the fact that the weighted power means μp(ABt(tA(− t)Bp)1/p, 1 ≤ p 2 , satisfy Audenaert's “in-betweenness” property for positive semidefinite matrices. We show that the “in-betweenness” property holds with respect to any unitarily invariant norm for = 1/2 and with respect to the Euclidean metric for = 1/4 . We also show that the only Kubo–Ando symmetric mean that satisfies the “in-betweenness” property with respect to any metric induced by a unitarily invariant norm is the arithmetic mean. Finally, for p=6  we give a counterexample to a conjecture by Audenaert regarding the “in-betweenness” property.