Let \(V\) be a nonnegative matrix. The nonnegative matrix factorization problem consists of finding nonnegative matrix factors \(W \in \mathbb{R}^{m,r}\) and \(H \in \mathbb{R}^{r,n}\) such that \(V \approx WH\). \textit{D. D. Lee} and \textit{H. S. Seung} [Unsupervised learning by convex and conic coding, Adv. Neural Inf. Process. Syst. 9, 515--521 (1997); Learning the parts of the objects by nonnegative matrix factorization, Nature 401, 788--791 (1999); Algorithms for nonnegative matrix factorization, Adv. Neural Inf. Process. Syst. 13 (2001)] proposed two algorithms, one of which finds nonnegative \(W\) and \(H\) such that \(\| V-WH\| _F\) is minimized. In this paper, the authors first study the case \(r=1\) about which they obtain a complete characterization of the solution. Next, they consider the case in which \(m=n\) and \(V\) is symmetric. They focus on questions concerning when the best approximate factorization results in the product \(WH\) being symmetric and on cases in which the best approximation cannot be a symmetric matrix. Finally, the authors show that the class of positive semidefinite symmetric nonnegative matrices \(V\) generated via a Soules basis admit for every \(1 \leq r \leq rank(V)\), a nonnegative factorization \(WH\) which coincides with the best approximation in the Frobenius norm to \(V\) in \(\mathbb{R}^{n,n}\) of rank not exceeding \(r\).