Matrix power means and the Karcher mean

DOI10.1016/j.jfa.2011.11.012zbMath1244.15014OpenAlexW2013011196MaRDI QIDQ665483

Yongdo Lim, Miklós Pálfia

Publication date: 5 March 2012

Published in: Journal of Functional Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jfa.2011.11.012

zbMATH Keywords

monotonicitygeometric meanconvex conepositive definite matrixpower meanRiemannian trace metricmatrix meanKarcher meanThompson metricmetric nonpositive curvatureRiemannian barycenter

Mathematics Subject Classification ID

Norms of matrices, numerical range, applications of functional analysis to matrix theory (15A60) Positive matrices and their generalizations; cones of matrices (15B48) Miscellaneous inequalities involving matrices (15A45) Operator means involving linear operators, shorted linear operators, etc. (47A64)

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Uses Software

• Matrix Means Toolbox

Cites Work

• Unnamed Item
• Monotonicity of the matrix geometric mean
• Completely positive mappings and mean matrices
• Operator log-convex functions and operator means
• Monotonic properties of the least squares mean
• Some inequalities for unitarily invariant norm
• Riemannian geometry and matrix geometric means
• Mean matrices and infinite divisibility
• Means of positive linear operators
• Convexity of the geodesic distance on spaces of positive operators
• On the exponential metric increasing property.
• Geometric means
• Computing the Karcher mean of symmetric positive definite matrices
• Multi-variable weighted geometric means of positive definite matrices
• Positive definite matrices
• Metric convexity of symmetric cones
• On Ando-Li-Mathias geometric mean equations
• On the averaging of symmetric positive-definite tensors
• The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order
• A general framework for extending means to higher orders
• An effective matrix geometric mean satisfying the Ando–Li–Mathias properties
• Matrix Analysis
• Riemannian center of mass and mollifier smoothing
• A Panoramic View of Riemannian Geometry
• A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
• Symmetric spaces with convex metrics
• Geometric Means in a Novel Vector Space Structure on Symmetric Positive‐Definite Matrices
• On Certain Contraction Mappings in a Partially Ordered Vector Space
• Compressions of infinite-dimensional bounded symmetric domains