Material for : Estimation of ( near ) Low - Rank Matrices with Noise and High - Dimensional Scaling

Part (a) of the claim was proved in Recht et al. [2]; we simply provide a proof here for completeness. We write the SVD as Θ∗ = UDV T , where U ∈ Rm1×m1 and V ∈ Rm2×m2 are orthogonal matrices, and D is the matrix formed by the singular values of Θ∗. Note that the matrices U r and V r are given by the first r columns of U and V respectively. We then define the matrix Γ = UTΔV ∈ Rm1×m2 , and write it in block form as