The Smallest Perturbation of a Submatrix which Lowers the Rank and Constrained Total Least Squares Problems

Let \[ T = [ {{array}{*{20}c} A & B \\ C & D \\ {array} } ]\] be a .partitioned rectangular matrix. We exhibit perturbations of D of smallest norm which lower the rank of T. We apply this result to generalize the total least squares problem $Sx = b$ when only a rectangular subset of the data $[S,b]$ is considered uncertain.