Theorem 5.1. Let A E Rmxn. Then there exist orthogonal matrices U E Rmxm and V E Rxn such that A = UEVI, (5.1) where E = 6 0 , S = diag(01, . ..,0,) E R'X, and o1 2 ... 2 0, > 0. More specifically, we have A = [UI U2] O O V (5.2) = UISVI . (5.3) The submatrix sizes are all determined by r (which must be