Let be the set of all 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 -tuple of via the Riemannian metric
where the , , are the (real and positive) eigenvalues of . The author points out that in the case that the 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.