Bearing in mind the relation between a diagram and its dual established
in Section 5.2, it is not surprising to see that all the de Bruijn
diagrams for k states form a chain of duals. In fact, 
, the null
string occupying its single node (see Figure 11), could begin
the chain.
.5em
Figure 11: Zero stage binary de Bruijn diagram.
In any event we can start the chain with 0.30em
0.30em
wherein the pattern of the factorization is fairly evident.
Suppose that 
is a 
unit matrix and that u is a
k-dimensional vector with unit components. Then evidently
From this it is immediately clear that the whole chain of binary
de Bruijn matrices has the single nonzero eigenvalue k belonging
to 
. Note that the factors in the Kronecker product cannot be
multiplied together because they are not conformable; the formulas
stand as written.