Projects - Cengage Learning

C-94 Appendix C
B
A
D
Figure 1
Communication links.
A
B
C
D
A B C D
0 1 1 0
1 0 1 0
§
¥
1 1 0 1
0 0 1 0
Figure 2
Communication matrix M.
To
d
PROJECT
A B C D
A 2 1 1 1
B 1 2 1 1
From d §
¥
C 1 1 3 0
D 1 1 0 1
Figure 3
The one-intermediary matrix
based on the spy network in
Figure 1.
C
Communications and Matrices
Matrices and matrix products are used in communication theory. Suppose that
four spies, A, B, C, and D, work for an intelligence agency under the following
restrictions: Spies A, B, and C all can communicate directly with one another.
Spy D, however, is super-secret and can communicate directly (back and forth)
only with spy C. These communication links are summarized in Figure 1.
The communication links between the four spies can also be specified by
means of the four-by-four communication matrix shown in Figure 2. We’ll call
this matrix M.
In the communication matrix the rows and columns are associated with the
spies as indicated by the red letters outside the matrix. An entry of 1 means that
the two spies associated with that matrix location can communicate directly.
For example, in the location row one, column two, there is a 1 because the corresponding
spies A and B can communicate directly (according to Figure 1). In
cases where two spies cannot communicate directly, 0 is used in the matrix. For
instance, there is a 0 in the location row two, column four because the two
spies B and D cannot communicate directly. There are four locations in the matrix
that are not covered by what we’ve said so far. These are the locations corresponding
to only one spy (i.e., a spy and him/herself):
row 1, column 1: spy A
row 2, column 2: spy B
By definition, we use a 0 in those locations.
Looking at Figure 1, we see that spy D cannot send a message directly to
spy A. However, spy D can get a message to A by using relays or intermediaries.
For instance, using one intermediary, the message from D to A can be
routed
D S C S A
Or using two intermediaries, the message can be routed
D S C S B S A
One could even have three intermediaries (perhaps to confuse the enemy) with
the route
D S C S B S C S A
row 3, column 3: spy C
row 4, column 4: spy D
We now display and explain a four-by-four matrix that shows how many
ways messages can be sent from one spy to another using only one intermediary.
See Figure 3.
As an example of how to interpret the entries in the one-intermediary matrix
in Figure 3, consider the 1 that appears in the second row, first column.
This tells us that there is exactly one way to send a message from B to A using
an intermediary. You can check that this is the case by looking at Figure 1; the
route is B S C S A. As another example, consider the 2 in the first row, first
column. This tells us that there are two ways that A can send a message to A
through one intermediary.As Figure 1 shows, the two routes are A S B S A and
A S C S A. (One possible reason why spy A might want to send a message to