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

MSC:
47A64 Operator means involving linear operators, shorted linear operators, etc.
26E60 Means
15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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