# zbMATH — the first resource for mathematics

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
Full Text: