Nonnegative Factorization of a Data Matrix as a Motivational Example for Basic Linear Algebra

  • Barak A. Pearlmutter
  • Helena ŠmigocEmail author
Part of the ICME-13 Monographs book series (ICME13Mo)


We present a motivating example for matrix multiplication based on factoring a data matrix. Traditionally, matrix multiplication is motivated by applications in physics: composing rigid transformations, scaling, sheering, etc. We present an engaging modern example which naturally motivates a variety of matrix manipulations, and a variety of different ways of viewing matrix multiplication. We exhibit a low-rank non-negative decomposition (NMF) of a “data matrix” whose entries are word frequencies across a corpus of documents. We then explore the meaning of the entries in the decomposition, find natural interpretations of intermediate quantities that arise in several different ways of writing the matrix product, and show the utility of various matrix operations. This example gives the students a glimpse of the power of an advanced linear algebraic technique used in modern data science.


Nonnegative matrix factorization (NMF) Topic modeling Data mining Matrix multiplication 


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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceMaynooth UniversityMaynoothIreland
  2. 2.School of Mathematics and StatisticsUCD DublinDublinIreland

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