The Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance

A ri gorous definition is given for th e concept of an "inte raction ma trix " (Zij) where i = 1 to In and j = 1 to n, in te rm s of two idempote nt matri ces A" and Bs of rank rand s, respec tively. It is then shown that the frequency distrib ution of the eigenvalues of (Z)(Z), depends only on rand s. App li ca tions are given to matri ces of residui arising from two-way data, ei the r by removing rowand/or colu mn-means, or by applying any number of sweeps of the "vacuum cleaner." Th e th eore ms are important in the theory of the analysis of two-way tables of nonadditive da ta.