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The Karcher mean of three variables and quadric surfaces
Journal of Mathematical Analysis and Applications ( IF 1.3 ) Pub Date : 2020-10-01 , DOI: 10.1016/j.jmaa.2020.124321
Hayoung Choi , Eduardo Ghiglioni , Yongdo Lim

Abstract The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean is problematic even for 2 × 2 triples. In this paper we study (1) the linear formula for the Karcher mean of 2 × 2 positive definite Hermitian matrices: Λ ( A , B , C ) = x A + y B + z C with nonnegative coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem, and (2) the quadric surface induced by the determinantal formula: det ⁡ ( A B C ) 1 3 = det ⁡ ( x A + y B + z C ) . We show that the solution set forms a simplex of dimension less than equal 2 and settle the first problem for linearly dependent case. A classification of the quadric surfaces from the linear form of Karcher means is presented in terms of linear (in)dependence of A , B , C : hyperboloid of two sheets, hyperbolic cylinder, and parallel planes.

中文翻译:

三变量和二次曲面的 Karcher 均值

摘要 Riemannian 或 Karcher 均值最近已成为对正定矩阵求平均和研究的重要工具。即使对于 2 × 2 三元组,找到 Karcher 均值的明确公式也是有问题的。在本文中,我们研究 (1) 2 × 2 正定 Hermitian 矩阵的 Karcher 均值的线性公式: Λ ( A , B , C ) = x A + y B + z C 具有非负系数,其中存在非负解决方案由 Sturm 的 SLLN 和 Holbrook 的无骰子定理保证,以及 (2) 由行列式得出的二次曲面: det ⁡ ( ABC ) 1 3 = det ⁡ ( x A + y B + z C ) 。我们证明了解集形成了一个维数小于等于 2 的单纯形,并解决了线性相关情况的第一个问题。
更新日期:2020-10-01
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