This paper studies the inferential theory for estimating low-rank matrices.
It also provides an inference method for the average treatment effect as an
application. We show that the least square estimation of eigenvectors following
the nuclear norm penalization attains the asymptotic normality. The key
contribution of our method is that it does not require sample splitting. In
addition, this paper allows dependent observation patterns and heterogeneous
observation probabilities. Empirically, we apply the proposed procedure to
estimating the impact of the presidential vote on allocating the U.S. federal
budget to the states