Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 27

other half filled (1). This makes it possible to re-label all elements, letting
the receptors as well as effectors run through labels 1 Æ m . Consequently,
a new matrix can be set up, an “action matrix”, which has precisely the same
property as our old connection matrix, with the only difference that the ele-
ments of the effector set—which define again the columns—are labeled
with the same indices as the receptor elements defining rows. Figs. 4a and
4b illustrate this transformation in a simple example.
The first advantage we may take of an action matrix is its possibility to
give us an answer to the question whether or not several action nets in
cascade may be replaced by a single action net; and if yes, what is the struc-
ture of this net?
The possibility of transforming a network into the form of an action-
matrix has considerable advantages, because an action matrix has the prop-
erties of an algebraic square matrix, so the whole machinery of matrix
manipulation that has been developed in this branch of mathematics can
be applied to our network structures. Of the many possibilities that can be
discussed in connection with matrix representation of networks, we shall
give two examples to illustrate the power of this method.
In algebra a square matrix A m of order m is a quadratic array of numbers
arranged precisely according the pattern of our connection matrix, or our
action matrix. The number found at the intersection of the i th row with the
j th column is called element a ij , which gives rise to another symbolism for
writing a matrix:
FIGURE 4. Network without feedback. Action net.