Identifying the components of a dataset is a fundamental problem in science and engineering. Non-negative matrix factorization (NMF) is a mathematical tool for performing this complicated task. Unfortunately, the non-negativity constraints prevents many well-developed linear algebra algorithms, such as the singular value decomposition, from being applicable to NMF. Despite this, several iterative algorithms for performing NMF have been recently developed. However, these algorithms tend to operate slowly and do not always identify the components which are most representative of the data.

In this thesis, an alternative algorithm for performing NMF is developed using the geometry of the problem. This algorithm is shown to be less computationally intensive than a popular existing algorithm. Moreover, the conditions under which the new algorithm can successfully identify the components of a dataset are explored and clearly defined.

The development of the new algorithm raises questions about the ill-posedness of the NMF problem. Indeed, numerical experiments illuminate the challenges and ambiguity of using NMF algorithms to analyze data. The results of these experiments, together with an analysis of the NMF framework, suggest that NMF is not sufficiently constrained to be applied successfully outside of a particular class of problems.