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Factorizations and geometric means of positive definite matrices. (English) Zbl 1251.15019

The author presents a new class of (metric) geometric means of positive definite matrices varying over Hermitian unitary matrices. He proves that each Hermitian unitary matrix induces a factorization of the cone \(\mathbb P_{m}\) of \(m\times m\) positive definite Hermitian matrices into geodesically convex subsets and a Hadamard metric structure on \(\mathbb P_{m}\). He also gives an explicit formula for the corresponding metric midpoint operation in terms of the geometric and spectral geometric means and shows that the resulting two-variable mean is different to the standard geometric mean. Some basic properties comparable to those of the geometric mean and its extensions to a finite number of positive definite matrices are also studied.

MSC:

15A23 Factorization of matrices
15B48 Positive matrices and their generalizations; cones of matrices
26E30 Non-Archimedean analysis
15B57 Hermitian, skew-Hermitian, and related matrices
15B10 Orthogonal matrices
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