  We solve the problem of best approximation by partial isometries of given
rank to an arbitrary rectangular matrix, when the distance is measured in any
unitarily invariant norm. In the case where the norm is strictly convex, we
parametrize all the solutions. In particular, this allow us to give a simple
necessary and sufficient condition for uniqueness. We then apply these results
to solve the global problem of approximation by partial isometries, and to
extend the notion of symmetric approximation of frames introduced in M. Frank,
V. Paulsen, T. Tiballi, Symmetric Approximation of frames and bases in Hilbert
Spaces, Trans. Amer. Math. Soc. 354 (2002), 777-793. In addition, we
characterize symmetric approximations of frames belonging to a prescribed
subspace.
