Leveraging this property, it is straightforward to prove the following lemma: Lemma 1.
while this statement is not true under the 2 norm, it is true under generalized KL divergence (Lee and Seung, 2001): generalized KL divergence preserves row and column sums: if M (κ) is the best rank κ approximation of M under gKL then the row sums and column sums of M (κ) and M are equal (Ho and Van Dooren, 2008).