  In this paper we study the low rank matrix completion problem using tools
from Schur complement. We give a sufficient and necessary condition such that
the completed matrix is globally unique with given data. We assume the observed
entries of the matrix follow a special "staircase" structure. Under this
assumption, the matrix completion problem is either globally unique or has
infinitely many solutions (thus excluding local uniqueness). In fact, the
uniqueness of the matrix completion problem totally depends on the rank of the
submatrices at the corners of the "staircase". The proof of the theorems make
extensive use of the Schur complement.
