# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Ando, T.; Li, C.K.; Mathias, R., Geometric means, Linear algebra appl., 385, 305-334, (2004) · Zbl 1063.47013 [2] Bhatia, R., Positive definite matrices, Princeton series in applied mathematics, (2007), Princeton University Press Princeton, NJ [3] Bauschke, H.H.; Goebel, R.; Lucet, Y.; Wang, X., The proximal average: basic theory, SIAM J. optim., 19, 766-785, (2008) · Zbl 1172.26003 [4] 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 [5] 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 [6] 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 [7] 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 [8] Lawson, J.D.; Lim, Y., The geometric Mean, matrices, metrics, and more, Amer. math. monthly, 108, 797-812, (2001) · Zbl 1040.15016 [9] Moakher, M., On the averaging of symmetric positive-definite tensors, J. elasticity, 82, 273-296, (2006) · Zbl 1094.74010 [10] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.