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A differential geometric approach to the geometric mean of symmetric positive-definite matrices. (English) Zbl 1079.47021
Let $\cal{P}(n)$ be the set of all $n\times n$ 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 $P_{i}\in \cal{P}(n)$ via the Riemannian metric $$\cal{G}(P_{1}, P_{2},..., P_{m}):= \mathop{\text{arg min}}_{P\in \cal{P}(n)} \sum_{k=1}^{m}\Vert \text{Log} (P_{k}^{-1}P) \Vert _{F}^{2}= \sum_{k=1}^{m}\sum_{i=1}^{n} (\log \lambda_{ki})^{2} ,$$ where the $\lambda_{ki}$, $i=1,2,...,n$, are the (real and positive) eigenvalues of $P_{k}^{-1}P$. The author points out that $\cal{G}(P_{1}, P_{2},..., P_{m})= (P_{1} P_{2}\cdots P_{m})^{1/n}$ 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 {\it W. N. Anderson Jr.} and {\it G. E. Trapp} [SIAM J. Appl. Math. 28, 60--71 (1975; Zbl 0295.47032)] and bye {\it W. Pusz} and {\it S. L. Woronowicz} [Rep. Math. Phys. 8, 159--170 (1975; Zbl 0327.46032)], and shows some properties of the geometric mean.

47A64Operator means, shorted operators, etc.
15B48Positive matrices and their generalizations; cones of matrices
15B57Hermitian, skew-Hermitian, and related matrices
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