Reduction of asymmetry by rank-one matrices
Gower has shown how to partition the sum of squares of an asymmetric matrix into independent parts associated with the symmetric and the skew-symmetric parts of the matrix; and has pointed out that asymmetry can be removed by subtracting certain unit-rank matrices, which improve the symmetry in equally contributing pairs. In the present paper, the definition of asymmetry implied in Gower's approach is analyzed and contrasted with the standard definition of asymmetry, entailing a different method of removing asymmetry. In particular, it is shown that, when the standard definition is adopted, only half the number of rank-one matrices is needed to remove asymmetry as is needed in Gower's approach. Some implications for graphical data analysis are discussed.
asymmetry graphical data analysis skew symmetry