Linear Algebra Seminar

Speaker: Trung Hoa Dinh

Title: On the monotonicity of weighted power means for matrices (this is joint work with Raluca Dumitru and Jose A. Franco from the University of North Florida)

Abstract: In this talk, we will discuss the "in-betweenness" property for positive semidefinite matrices. We will provide an alternate proof of the fact that the weighted power means

$\mu_p(A,B,t)=(tA^p+(1-t)B^p)^{1/p},\phantom{X} 1\leq p \leq 2$ has the "in-betweenness" property. We show that the "in-betweenness" property holds with respect to any unitarily invariant norm for $p=1/2$ and with respect to the Euclidean metric for $p=1/4$. We will 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. Some open problems will be discussed.