10000 rows and 10 columns and a coefficients matrix with 10 rows and 500 columns. The product of and is a matrix with 10000 rows and 500 columns, the same shape as the input matrix and, if the factorization worked, it is a reasonable approximation to the input matrix . From the treatment of matrix multiplication above it follows that each column in the product matrix is a linear combination of the 10 column vectors in the features matrix with coefficients supplied by the coefficients matrix . This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in ) where each feature is weighted by the feature's cell value from the document's column in . Clustering property NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data . More specifically, the approximation of by is achieved by finding and that minimize the error function (using the Frobenius norm) subject to , If we furthermore impose an orthogonality constraint on , i.e. , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed gives the cluster membership, i.e., if for all i ≠ k, this suggests that the input data belongs to -th cluster. The computed gives the cluster centroids, i.e., the -th column gives the cluster centroid of -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. Clustering is the main objective of most data mining applications of NMF. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document