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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 CD is the metric midpoint of the of arithmetic mean A=1 2(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 ω(0,1) m with ω 1 =1 and positive definite matrices A 1 ,,A m with A=(A 1 ,,A m ) they introduce the weighted AH-mean to be the matrix geometric mean of the weighted arithmetic and harmonic means: (ω;A):= i ω i A i i ω i A i -1 -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:
15B48Positive matrices and their generalizations; cones of matrices
47A64Operator means, shorted operators, etc.
26E60Means
References:
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