I am trying to understand the effect of noise data on matrix. Here is the question,
A is a rank r matrix of dimensions n x d and its SVD is given below
$$A=USV^T$$
Upon addition of a noise matrix E, I get a new matrix of rank r1 (r1 > r) as shown below $$A_1 = A + E$$
$$A_1=U_1S_1V_1^T$$
Further, I have a third matrix $$\hat{A}=U_1SV_1^T$$ which is generated by only considering the top r entries of the matrix A1.
How can I prove the following:
$$\lVert \hat{A} - A \rVert _ F <= √8r * \lVert E \rVert_2$$
Any help is appreciated. Thanks!