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.
