RC and CR factorizations

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.

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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.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx