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

Let \ T = [ array{*{20c} 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.