Abstract
Given two positive semidefinite Hermitian matrices A and B there are natural definitions for their arithmetic and harmonic means. In this work we consider the following question: Given positive semidefinite matrices C and D, when do there exist positive semidefinite matrices A and B such that C is the arithmetic mean of A and B and D is the harmonic mean of A and B. Uniqueness questions are also answered. Similar questions are answered concerning the geometric mean.
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