  In this paper the geometric mean of partial positive definite matrices with
missing entries is considered. The weighted geometric mean of two sets of
positive matrices is defined, and we show whether such a geometric mean holds
certain properties which the weighted geometric mean of two positive definite
matrices satisfies. Additionally, counterexamples demonstrate that certain
properties do not hold. A Loewner order on partial Hermitian matrices is also
defined. The known results for the maximum determinant positive completion are
developed with an integral representation, and the results are applied to the
weighted geometric mean of two partial positive definite matrices with missing
entries. Moreover, a relationship between two positive definite completions is
established with respect to their determinants, showing relationship between
their entropy for a zero-mean,multivariate Gaussian distribution. Computational
results as well as one application are shown.
