Geometric means

Geometric Means

We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and that our geometric mean generalizes many inequalities satisfied by the geometric mean of two positive semidefinite matrices. We prove some new properties of the geometric mean of ...

Constructing matrix geometric means

In this paper, we analyze the process of “assembling” new matrix geometric means from existing ones, and show what new means can be found, and what cannot be done because of group-theoretical obstructions. We show that for n = 4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that for n > 4 the existing strategies of composing matrix means and taki...

Ela Constructing Matrix Geometric Means

In this paper, we analyze the process of “assembling” new matrix geometric means from existing ones, through function composition or limit processes. We show that for n = 4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that for n > 4 the existing proving strategies cannot provide a mean computationally simpler than the existing ones.

Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

Reversed geometric programs treated by harmonic means

A "posynomial" is a (generalized) polynomial with arbitrary real exponents, but positive coefficients and positive independent variables. Each posynomial program in which a posynomial is to be minimized subject to only inequality posynomial constraints is termed a "reversed geometric program". The study of each reversed geometric program is reduced to the study of a corresponding family of appr...