Matrix power means and the Karcher mean

@article{Lim2012MatrixPM,
  title={Matrix power means and the Karcher mean},
  author={Y. Lim and Mikl{\'o}s P{\'a}lfia},
  journal={Journal of Functional Analysis},
  year={2012},
  volume={262},
  pages={1498-1514}
}
We define a new family of matrix means {Pt(ω;A)}t∈[−1,1], where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these means except t≠0 arises as a unique positive definite solution of a non-linear matrix equation, satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means. The main result is that the Karcher mean coincides with the limit of power… Expand
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  • J. Lawson, Y. Lim
  • Computer Science, Medicine
  • Proceedings of the National Academy of Sciences
  • 2013
TLDR
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References

SHOWING 1-10 OF 33 REFERENCES
Multi-variable weighted geometric means of positive definite matrices
Abstract We define a family of weighted geometric means { G ( t ; ω ; A ) } t ∈ [ 0 , 1 ] n where ω and A vary over all positive probability vectors in R n and n-tuples of positive definite matricesExpand
Classification of affine matrix means
In this article we find all possible matrix means which are points of geodesics of affinely connected manifolds. We characterize certain properties of these manifolds, decide whether they areExpand
Positive Definite Matrices
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive realExpand
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices
TLDR
This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. Expand
Completely positive mappings and mean matrices
Abstract Some functions f : R + → R + induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define aExpand
The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order
The Riemannian mean on the convex cone of positive definite matrices is a kind of geometric mean of n -matrices which is an extension of the geometric mean of two-matrices. In this paper, we deriveExpand
A general framework for extending means to higher orders
Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It hasExpand
ON CERTAIN CONTRACTION MAPPINGS IN A PARTIALLY ORDERED VECTOR SPACE
G. Birkhoff [1] and H. Samelson [4] have shown that a means of solving problems concerning the existence and uniqueness of eigenvectors of positive operators is given by introducing a suitable metricExpand
Riemannian geometry and matrix geometric means
The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authorsExpand
Monotonicity of the matrix geometric mean
An attractive candidate for the geometric mean of m positive definite matrices A1, . . . , Am is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in theExpand
...
1
2
3
4
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