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A geometric mean for symmetric spaces of noncompact type. (English) Zbl 1331.15011

Let \(A,B\in\mathbb{C}^{n\times n}\) be Hermitian positive definite, let \(\mathbb{P}_n\) denote the set of all such matrices, and let \(t,s\in\mathbb{R}\) satisfy \(0\leq t\leq 1\) and \(s>0\). The \(t\)-geometric mean of \(A\) and \(B\) is defined by \[ A\,\#_t\,B=A^\frac{1}{2}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^tA^\frac{1}{2}. \] The set \(\mathbb{P}_n\) can be equipped with a suitable Riemannian metric so that the curve \(\gamma(t)=A\,\#_t\,B\), \(0\leq t\leq 1\), is the unique geodesic joining \(A\) and \(B\); see [R. Bhatia, Positive definite matrices. Princeton, NJ: Princeton University Press (2007; Zbl 1133.15017)].
Denote by \(\lambda(\cdot)\) the vector of ordered eigenvalues. Then \[ \lambda(A\,\#_t\,B)\prec_{\log}\lambda(e^{(1-t)\log{A}+t\log{B}})\prec_{\log} \lambda(B^\frac{ts}{2}A^{(1-t)s}B^\frac{ts}{2})^\frac{1}{s}= \lambda(A^{(1-t)s}B^{ts})^\frac{1}{s}, \] where \(\prec_{\log}\) stands for log majorization, see, e.g. [R. Bhatia and P. Grover, Linear Algebra Appl. 437, No. 2, 726–733 (2012; Zbl 1252.15023)].
The authors extend these interesting results to symmetric spaces of noncompact type. The space of matrices in \(\mathbb{P}_n\) with determinant \(1\) is such a space, and the relation \(\prec_{\log}\) is Kostant’s preorder there.

MSC:

15A45 Miscellaneous inequalities involving matrices
15B48 Positive matrices and their generalizations; cones of matrices
53C35 Differential geometry of symmetric spaces
26E60 Means
53C22 Geodesics in global differential geometry
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