  Positive semidefinite Hermitian matrices that are not fully specified can be
completed provided their underlying graph is chordal. If the matrix is positive
definite the completion can be uniquely characterized as the matrix that
maximizes the determinant, or as the matrix whose inverse has zeroes in those
places that were undetermined in the original matrix. This paper extends these
uniqueness results to the case of semidefinite matrices. Because the
determinant vanishes for singular matrices, and because the inverse does not
exist, we introduce a generalized determinant and use generalized inverses to
formulate equivalent characterizations in the semidefinite case. For a class of
matrices that are singular but of maximal rank unique characterizations can be
given, just as in the positive definite case.
