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The matrix geometric mean of parameterized, weighted arithmetic and harmonic means. (English) Zbl 1222.15036
For positive definite matrices $$C$$ and $$D$$, the matrix geometric mean $$C \sharp D$$ is the metric midpoint of the of arithmetic mean $$A = \frac12(C + D)$$ and the harmonic mean $$H = 2(C^{-1} + D^{-1})^{-1}$$ for the trace metric. The authors consider the more general construction of taking the geometric mean of the weighted $$n$$-variable arithmetic and harmonic means. More precisely, for $$\omega \in (0, 1)^m$$ with $$\|\omega\|_1=1$$ and positive definite matrices $$A_1,\dots,A_m$$ with $$A = (A_1,\dots,A_m)$$ they introduce the weighted $$A\sharp H$$-mean to be the matrix geometric mean of the weighted arithmetic and harmonic means: $$\mathcal L(\omega;A) := \left( \sum_i \omega_i A_i\right) \sharp \left( \sum_i \omega_i A_i^{-1}\right)^{-1}$$. Many properties of this weighted mean are presented, and it is interpreted via the Kullback-Leibler divergence from probability theory and information theory.

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 47A64 Operator means involving linear operators, shorted linear operators, etc. 26E60 Means
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##### References:
  Ando, T.; Li, C.K.; Mathias, R., Geometric means, Linear algebra appl., 385, 305-334, (2004) · Zbl 1063.47013  Bhatia, R., Positive definite matrices, Princeton series in applied mathematics, (2007), Princeton University Press Princeton, NJ  Bauschke, H.H.; Goebel, R.; Lucet, Y.; Wang, X., The proximal average: basic theory, SIAM J. optim., 19, 766-785, (2008) · Zbl 1172.26003  Bauschke, H.H.; Moffat, S.M.; Wang, X., The resolvent average for positive semidefinite matrices, Linear algebra appl., 432, 1757-1771, (2010) · Zbl 1191.15024  Bauschke, H.H.; Borwein, J.M., Legendre functions and the method of random Bregman projections, J. convex anal., 4, 27-67, (1997) · Zbl 0894.49019  Bini, D.; Meini, B.; Poloni, F., An effective matrix geometric Mean satisfying the ando – li – mathias properties, Math. comp., 79, 437-452, (2010) · Zbl 1194.65065  Chergui, M.; Leazizi, F.; Raïssouli, M., Arithmetic – geometric – harmonic Mean of three positive operators, J. inequalities pure appl. math., 10, 4, (2009) · Zbl 1185.65092  Lawson, J.D.; Lim, Y., The geometric Mean, matrices, metrics, and more, Amer. math. monthly, 108, 797-812, (2001) · Zbl 1040.15016  Moakher, M., On the averaging of symmetric positive-definite tensors, J. elasticity, 82, 273-296, (2006) · Zbl 1094.74010  M. Moakher, P.G. Batchelor, Symmetric Positive-definite Matrices: From Geometry to Applications and Visualization, Math. Vis., vol. 452, Springer, Berlin, 2006, pp. 285-298.
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