  The Schur-Horn theorem is a classical result in matrix analysis which
characterizes the existence of positive semidefinite matrices with a given
diagonal and spectrum. In recent years, this theorem has been used to
characterize the existence of finite frames whose elements have given lengths
and whose frame operator has a given spectrum. We provide a new generalization
of the Schur-Horn theorem which characterizes the spectra of all possible
finite frame completions. That is, we characterize the spectra of the frame
operators of the finite frames obtained by adding new vectors of given lengths
to an existing frame. We then exploit this characterization to give a new and
simple algorithm for computing the optimal such completion.
