scholarly journals Multi-variable weighted geometric means of positive definite matrices

2011 ◽  
Vol 435 (2) ◽  
pp. 307-322 ◽  
Author(s):  
Hosoo Lee ◽  
Yongdo Lim ◽  
Takeaki Yamazaki
Keyword(s):  
Positive Definite ◽  
Positive Definite Matrices ◽  
Geometric Means

Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices

2004 ◽  
Vol 56 (4) ◽  
pp. 776-793 ◽  
Author(s):  
Yongdo Lim
Keyword(s):  
Best Approximation ◽  
Positive Definite ◽  
Explicit Calculation ◽  
Hadamard Manifold ◽  
Minimal Distance ◽  
Positive Definite Matrices ◽  
Geometric Means ◽  
Geodesic Submanifolds ◽  
The Best Approximation ◽  
Schur Complements

AbstractWe explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold Sym(n, ℝ)++ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold Sym(p, ℝ)++ × Sym(q, ℝ)++ block diagonally embedded in Sym(n, ℝ)++ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when p ≤ 2 or q ≤ 2.

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