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Geometric means. (English) Zbl 1063.47013

Let G(A,B) be the geometric mean of two n×n positive semidefinite matrices A and B. The authors extend the definition of G to any number of n×n positive semidefinite matrices inductively. Suppose that for some k2, the geometric mean G(A 1 ,A 2 ,,A k ) of any k positive semidefinite matrices A 1 ,A 2 ,,A k has been defined. Let A=(A 1 ,A 2 ,,A k ,A k+1 ) be a (k+1)-tuple of n×n positive semidefinite matrices. Define T(A)(G((A i ) i1 ),G((A i ) i2 ),,G((A i ) ik+1 )).

The authors show that the sequence (T r (A)) r=1 has a limit of the form (A ˜,,A ˜) and define G(A 1 ,A 2 ,,A k ,A k+1 )=A ˜. The definition given here is the only one in the literature that has the properties that one would expect from a geometric mean. The authors also prove some new properties of the geometric mean of two matrices, and give some simple computational formulae related to them for 2×2 matrices.


MSC:
47A64Operator means, shorted operators, etc.
47A63Operator inequalities
15A45Miscellaneous inequalities involving matrices