Metric and Spectral Geometric Means on Symmetric Cones
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In a development of ecient primal-dual interior-points algorithms for self-scaled convex programming problems, one of the important properties of such cones isthe existence and uniqueness of “scaling points”. In this paper through the identicationof scaling points with the notion of “(metric) geometric means” on symmetric cones, weextend several well-known matrix inequalities (the classical L¨owner-Heinz inequality, Andoinequality, Jensen inequality, Furuta inequality) to symmetric cones. We also develop atheory of spectral geometric means on symmetric cones which has recently appeared inmatrix theory and in the linear monotone complementarity problem for domains associatedto symmetric cones. We derive Nesterov-Todd inequality using the spectral property ofspectral geometric means on symmetric cones.
Abstract
Abstract. In a development of efficient primal-dual interior-points algorithms for selfscaled convex programming problems, one of the important properties of such cones is the existence and uniqueness of “scaling points”. In this paper through the identification of scaling points with the notion of “(metric) geometric means” on symmetric cones, we extend several well-known matrix inequalities (the classical L¨owner-Heinz inequality, Ando inequality, Jensen inequality, Furuta inequality) to symmetric cones. We also develop a theory of spectral geometric means on symmetric cones which has recently appeared in matrix theory and in the linear monotone complementarity problem for domains associated to symmetric cones. We derive Nesterov-Todd inequality using the spectral property of spectral geometric means on symmetric cones. 1. Introduction
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