On an eigenvalue property relevant in correspondence analysis and related methods

Abstract

In correspondence analysis as well as in related methods such as dual scaling and homogeneity analysis, one encounters singular values of a matrix Q = D r −1/2 FD c −1/2, where F is an n× p nonnegative data matrix and D r and D c are diagonal matrices implicitly defined by the equations D r 1 n= F1 p and D c 1 p= F ′ 1 n. ( 1 i denotes the summation vector of order i.) An important and often cited property of these singular values is that they lie in the [0,1] interval. In this paper, this property will first be examined in the context of the aforementioned, mathematically equivalent, statistical methods. It will become apparent that for proving the property, knowledge of the method at hand, i.e. dual scaling, correspondence analysis or homogeneity analysis, is essential. We shall then show, by using the general matrix Q , that the result follows by elementary matrix algebra due to the nonnegativity of F and the scalings imposed by the diagonal matrices D r and D c .