PROXIMITY AND ORDER IN ASYMMETRIC PAIRED COMPARISON DATA By

In a square asymmetric matrix, the relationships among objects in the lower triangular half-matrix, differ from the relationships among the same objects in the upper triangular half. Square, asymmetric matrices can arise in similarity and preference data, when the direction of comparison is important. An asymmetric matrix can be rendered symmetric by averaging corresponding entries above and below the main diagonal. The difference between the original and the symmetric matrix is purely asymmetric, or skew-symmetric. The symmetric and skew-symmetric pans are orthogonal. An eigenvector-eigenvalue decomposition analyses the asymmetries into rank 2 skew-symmetric matrices, having an optimum least squares fit to the asymmetries (Gower, 1977)