4.2 The Symmetrical Converging structure
The general form of
in the case of symmetry is
.
It need not be that all
pairs of measurement are identical, but there still exists structures wherein the p points can't be divided into two groups. Figure 7 summarizes all the possible symmetry and nonsymmetry structures with their corresponding limits for p=3 and p=4. The number of possible limits is an increasing function of p.
In general there are only three types of columns in a converged pattern matrix:

columns with only plus and minus ones in a rank1 matrix, patterns 3(3), 4(5), and 4(6) for example;

columns with summation of elements equal to zero. (this type of matrix can be further divided into two subtypes one for symmetry, patterns 3(1) and 4(1) for instance; the other for a circular matrix, patterns 3(1), 4(2) and part of 4(3));

columns for symmetry
center(s) with zeroes to all other points, column B in pattern 3(2), column D in pattern 4(3), and columns B/C in pattern 4(4) for example. Section 5.4 studies the possible applications of using the symmetry patterns to study the crystallographic structure.
Figure 7. All possible Symmetry and NonSymmetry Structures with the
Corresponding Converged Pattern Matrices for p=3 and 4.
We usually do not encounter proximity matrices with a perfect symmetric structure in a medical study of the sort looked at here. One can search for a proximity matrix with a symmetric structure in physics, chemistry or molecular biology.
