  The transverse peculiar velocities caused by the mass distribution of
large-scale structure provide a test of the theoretical matter power spectrum
and the cosmological parameters that contribute to its shape. Typically, the
matter density distribution of the nearby Universe is measured through redshift
or line-of-sight peculiar velocity surveys. However, both methods require
model-dependent distance measures to place the galaxies or to differentiate
peculiar velocity from the Hubble expansion. In this paper, we use the
correlated proper motions of galaxy pairs from the VLBA Extragalactic Proper
Motion Catalog to place limits on the transverse peculiar velocity of galaxy
pairs with comoving separations <1500 Mpc without a reliance on precise
distance measurements. The relative proper motions of galaxy pairs across the
line of sight can be directly translated into relative peculiar velocities
because no proper motion will occur in a homogeneous expansion. We place a 3
sigma limit on the relative proper motion of pairs with comoving separations <
100 Mpc of -17.4 microas/yr < thetadot / sin theta < 19.8 microas/yr. We also
confirm that large-separation objects (> 200 Mpc) are consistent with pure
Hubble expansion to within ~ 5.3 microas/yr (1 sigma). Finally, we predict that
Gaia end-of-mission proper motions will be able to significantly detect the
mass distribution of large-scale structure on length scales < 25 Mpc. This
future detection will allow a test of the shape of the theoretical mass power
spectrum without a reliance on precise distance measurements.
