A geometric mean for symmetric spaces of noncompact type (Q2927935)

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 NEWLINE\[NEWLINE A\,\#_t\,B=A^\frac{1}{2}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^tA^\frac{1}{2}. NEWLINE\]NEWLINE 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 [\textit{R. Bhatia}, Positive definite matrices. Princeton, NJ: Princeton University Press (2007; Zbl 1133.15017)].NEWLINENEWLINEDenote by \(\lambda(\cdot)\) the vector of ordered eigenvalues. Then NEWLINE\[NEWLINE \lambda(A\,\#_t\,B)\prec_{\log}\lambda(e^{(1-t)\log{A}+t\log{B}})\prec_{\log} \lambda(B^\frac{ts}{2}A^{(1-t)s}B^\frac{ts}{2})^\frac{1}{s}= \lambda(A^{(1-t)s}B^{ts})^\frac{1}{s}, NEWLINE\]NEWLINE where \(\prec_{\log}\) stands for log majorization, see, e.g. [\textit{R. Bhatia} and \textit{P. Grover}, Linear Algebra Appl. 437, No. 2, 726--733 (2012; Zbl 1252.15023)].NEWLINENEWLINEThe 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.