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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
Appendix C C-95 him/herself through an intermediary would be to test the integrity of that intermediary.)As a final example in interpreting the matrix in Figure 3, consider the 0 in the bottom row. This says that there is no way for D to send a message to C using exactly one intermediary. (Check this in Figure 1.) There is a surprising connection between the communication matrix M in Figure 2 and the one-intermediary matrix in Figure 3. If you compute the matrix product of M with itself (this is abbreviated M 2 ), you get the oneintermediary matrix in Figure 3.You should verify this now for yourself. That is, either by hand or with a graphing utility, compute M 2 and check that the product is the same as the matrix in Figure 3. It can be shown that this situation holds in general. F Figure 4 H E To d G A B C D A 2 ? ? ? B ? ? ? ? From d § ¥ C ? ? ? ? D ? 1 3 ? Figure 5 Two-intermediary matrix for the spy network in Figure 1. A B C D E A B C D E 0 1 1 1 1 1 0 1 0 0 • 1 1 0 1 0μ 1 0 1 0 1 1 0 0 1 0 Figure 6 Communication matrix M for a network with five spies. Exercises 1. Consider the communication links shown in Figure 4. (a) Specify the communication matrix M for this system. (b) By referring to Figure 4, work out the one-intermediary matrix for this system. Then check your answer by computing M 2 and comparing the results. 2. Return to the spy network that we considered in Figure 1. (a) Work out the entries for the following two-intermediary matrix in Figure 5. As examples, three entries are already filled in. The 2 in the first row, first column indicates that there are two ways to send a message from A to A using two intermediaries. The routes are A S B S C S A and A S C S B S A. As another example, the 1 in the fourth row, second column indicates that there is only one way to send a message from D to B using two intermediaries. As you can verify in Figure 1, the route is D S C S A S B. Finally, consider the 3 in the fourth row, third column. This indicates that there are three ways to send a message from D to C using two intermediaries. The three routes are D S C S AS C, D S C S B S C, and D S C S D S C. In these routes from D to C, note that D and C themselves can be used as intermediaries. (b) Let M be the communication matrix given in Figure 2. It can be shown that the matrix M 3 [M(M 2 )] equals the two-intermediary matrix for this system. Compute M 3 , either by hand or with a graphing utility, and then use the result to check the entries in your answer for part (a). 3. Consider a network of five spies, A, B, C, D, and E with the communication matrix M shown in Figure 6. (a) Draw a diagram, of the type shown in Figure 1, showing the communication links. (b) Using the diagram from part (a): How many one-intermediary routes can you find for sending a message from D to A? How many twointermediary routes can you find for sending a message from D to A? (c) Use a graphing utility to compute M 2 and M 3 , and thereby check to see whether you missed any routes in part (b).
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APPENDIX C Title and Page Number Pr
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Appendix C C-3 MINI PROJECT Discuss
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Appendix C C-5 MINI PROJECT Thinkin
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Appendix C C-7 1. equilateral trian
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Appendix C C-9 MINI PROJECT Flying
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Appendix C C-11 MINI PROJECT An Ine
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Appendix C C-13 (a) The two data po
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Appendix C C-15 and http://www.svob
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Appendix C C-17 For Figure D we can
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Appendix C C-19 MINI PROJECT A Grap
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Appendix C C-21 MINI PROJECT Who Ar
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Appendix C C-23 MINI PROJECT 1 How
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Appendix C C-25 either the quadrati
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Appendix C C-27 PROJECT The Least-S
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Appendix C C-29 Then, for any line
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Appendix C C-31 PROJECT Finding Som
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Appendix C C-33 3. When a person co
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Appendix C C-35 2. Use part (b) of
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Appendix C C-37 PROJECT Coffee Temp
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Appendix C C-39 MINI PROJECT More C
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Appendix C C-41 The calculations co
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Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
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Page 63 and 64: Appendix C C-63 denote the angles o
Page 65 and 66: Appendix C C-65 inverse trigonometr
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Page 73 and 74: Appendix C C-73 vertically downward
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Page 77 and 78: Appendix C C-77 Exercise 8 We outli
Page 79 and 80: Appendix C C-79 y y=2x+5 Îy=2 A (1
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Page 89 and 90: Appendix C C-89 2. An altitude of a
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