A note on the eigensystem of the covariance matrix of dichotomous Guttman items

Abstract

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We consider the sample 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).

Keywords

• Principal Component Analysis
• Rasch model
• Eigenvalues
• eigenvectors
• Guttman scale
• Dichotomous items