title: A Note on the Eigensystem of the Covariance Matrix of Dichotomous Guttman Items.
abstract: We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions. We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).
authors.affiliation: Department of Psychological Sciences, University of Missouri Columbia, MO, USA.
authors.forename: Clintin P
authors.initials: CP
authors.lastname: Davis-Stober
authors.affiliation: Department of Mathematics, Université Libre de Bruxelles Brussels, Belgium.
authors.forename: Jean-Paul
authors.initials: JP
authors.lastname: Doignon
authors.affiliation: Universität Osnabrück Osnabrück, Germany.
authors.forename: Reinhard
authors.initials: R
authors.lastname: Suck
keywords: Guttman scale; Rasch model; dichotomous items; eigenvalues; eigenvectors; principal component analysis