Duals and homomorphism

 

The head matrix is column stochastic, because only one node can be the head of a link; of course several links can terminate at the same node. Similarly the tail matrix is row stochastic, there being only one tail to a given link. Strictly stochastic means that elements sum up to one, but the only possibility for integer elements in such a sum are zeroes and ones; just one single one may be present.

Dedicating the letters R and C to such matrices is more mnemonic than using and ; passing to the dual matrix then amounts to finding a CR factorization

 

then writing

 

The second dual would result from writing

 

with

 

and so on.

Note that if Equation 5 is multiplied on the right by R, the result

 

asserts that is homomorphic to M. So also is the second dual (and all the rest, for that matter), inasmuch as

The matrix R which generates the homomorphism from the dual to its graph associates each link with its terminal node.

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