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A differential geometric approach to the geometric mean of symmetric positive-definite matrices. (English) Zbl 1079.47021

Let 𝒫(n) be the set of all n×n real positive matrices. In this paper, the author defines some matrix means in terms of some metrics. Specifically, the author defines the geometric mean of an m-tuple (P 1 ,P 2 ,···,P m ) of P i 𝒫(n) via the Riemannian metric

𝒢(P 1 ,P 2 ,···,P m ):=argmin P𝒫(n) k=1 m Log(P k -1 P) F 2 = k=1 m i=1 n (logλ ki ) 2 ,

where the λ ki , i=1,2,···,n, are the (real and positive) eigenvalues of P k -1 P. The author points out that 𝒢(P 1 ,P 2 ,···,P m )=(P 1 P 2 P m ) 1/n in the case that the P i commute with each other. Next, the author shows that this geometric mean is the same as the mean defined by W. N. Anderson Jr. and G. E. Trapp [SIAM J. Appl. Math. 28, 60–71 (1975; Zbl 0295.47032)] and bye W. Pusz and S. L. Woronowicz [Rep. Math. Phys. 8, 159–170 (1975; Zbl 0327.46032)], and shows some properties of the geometric mean.


MSC:
47A64Operator means, shorted operators, etc.
26E60Means
15A48Positive matrices and their generalizations (MSC2000)
15A57Other types of matrices (MSC2000)