A rigorous definition is given for the concept of an interaction matrix (Z* lij*s) where i=1 to m and j=1 to n, in terms of two idempotent matrices A*lr*s and B*ls*s of rank r and s, respectively. It is then shown that the frequency distribution of the eigenvalues of (Z)(Z)' depends only on r and s. Applications are given to matrices of residuals arising from two-way data, either by removing row and/or column-means, or by applying any number of sweeps of the vacuum cleaner. The theorems are important in the theory of the analysis of two-way tables of nonadditive data.