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Singular Vectors' Subtle Secrets

Social scientists use adjacency tables to discover influence networks within and among groups. Building on work by Moler and Morrison, we use ordered pairs from the components of the first and second singular vectors of adjacency matrices as tools to distinguish these groups and to identify particularly strong or weak individuals in them. - See more at: http://www.maa.org/publications/periodicals/college-mathematics-journal/...

Identifier: 
http://www.jstor.org/stable/pdfplus/10.4169/college.math.j.42.2.086.pdf
Subject: 
Rating: 
Average: 4.5 (2 votes)
Creator(s): 
David James, Michael Lachance, and Joan Remski
Cataloger: 
Daniel Drucker
Publisher: 
College Math. Journal, Vol. 42, No. 2 (March 2011), pp. 86–95.
Rights: 
David James, Michael Lachance, and Joan Remski

Comments

drucker@math.wayne.edu's picture

The singular value decomposition of a matrix can be used to obtain an expansion of a matrix as a sum of rank one matrices weighted by the singular values. Roughly speaking, the terms corresponding to smaller singular values are less important. This article sheds light on the meaning of the two most important terms of the rank one expansion in the case where the matrix is an adjacency matrix. It is shown that the expansion helps to find groups or cliques and to identify whether individuals are strong or weak members of their cliques. The applications considered are varied: textual analysis, the tendency of Supreme Court justices to agree or disagree with one another, and the opinions that monks in a New England monastery have of one another. Very interesting article, even if its focus is not of central importance in most linear algebra courses.

meade's picture

Nicely written summary of the SVD, illustrated with 3 non-standard (but easily understood by students). This paper could be used as a foundation for student projects in a linear algebra course or for a capstone course.

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