2.2. DECOMPOSITION INTO STRONG COMPONENTS
47
and the partition of the associated polynomial expression is illustrated. This
section also gives an idea how quantitative and qualitative information can
be extracted from adjacency matrices if they are decomposed into irreducible
submatrices.
Note that the following (0,1)-matrix A does not contain artificial data
but describes the preferences of Subject 1 in Session 1 of Experiment 2B (see
Chapter 3).
0 110 1001000 1
0000000 00 0 0 1
O10111111101
110010010001
0 100000000 0 1
110110111001
A =
110110010001
0 10 0 1000000 0
110110110001
110111111001
11111111110
0 0 0 0 0 0 0 1 0 0 0 0
The index of the rows and columns refers to the number of the twelve lotteries
as listed in Table B.10. The subject made choices between all possible pairs
of lotteries in a forced choice pair comparison resulting in a tournament. The
entry a;; in matrix A is 1 if the subject preferred the Lottery i in row i over
the Lottery j in columnj and 0 otherwise. By simultaneous permutations of
the rows and columns according to their row sums, the following Frobenius
normal form of the tournament can be found.
1 1
111111
1
10
1 1 1 1
1 0
0 1111
0
1 11 1
0 011111
1 1 1 1
0001111
1
0 0 0 0 1 1 1
1 1 1
0 0 0 0 0 1 1
1 1 11
PAPT =
0 00000 0
1111
10000 10
1 1 11
0 1 0 11
0 0 000 0 0
0
0 0 1 1
0 0 0 0 0 0 0
00 000 0 0
100 0
00 00000000 10
The irreducible submatrices along the diagonal, which correspond to the
strong components of the tournament, are indicated by square brackets.