If X is of rank r, then Σ is also of rank r. Let Σ k , where k < r, be the diagonal matrix formed from the top k singular values, and let U k and V k be the matrices produced by selecting the corresponding columns from U and V. The matrix U k Σ k V T k is the matrix of rank k that best approximates the original matrix X, in the sense that it minimizes the approximation errors.
,X = U k Σ k V T k minimizes X − X F over all matricesX of rank k, where .