Remedies for Misapplications of Sylvester’s Criterion: A Pedagogic Illustration

Abstract

Sylvester’s criterion, which verifies the positive definiteness of any real symmetric matrix by examining the signs of all leading principal minors, is an excellent analytical tool. However, as noted by several authors in various academic fields, misapplications of the tool include its unjustified use for non-symmetric matrices and its unjustified extension for verifying the positive semidefiniteness of matrices. As remedies are available, this paper provides a pedagogic illustration that connects the corresponding tests for the positive definiteness and the positive semidefiniteness of a matrix to the underlying concepts. Drawing on standard materials in linear algebra, this paper uses self-contained Excel worksheets to illustrate the concepts involved and to generate matrices that are suitable for use in courses covering Sylvester’s criterion. This paper also suggests the use of some Excel-based exercises for students, as an alternative approach to cover the same topic.