  We consider in this paper two different types of the weighted geometric means
of positive definite operators. We show the component-wise bijection of these
geometric means and give a geometric property of the spectral geometric mean as
a metric midpoint. Moreover, several interesting inequalities related with the
geometric means of positive definite operators will be shown. We also see the
meaning of weighted geometric means in the gyrogroup structure with finite
dimension and find the formulas of weighted geometric means of 2-by-2 positive
definite matrices and density matrices.
