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

Exercise 2.3.10. (5 points) Which of the matrices below tend to a limiting steady state vector, and if so, what is it? We will take three state vectors, (1,0,0), (0,1,0), and (0,0,1) and see what happens after many stages -- say n=10, 100. Stacking them together simply gives the identity matrix, and since 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, all we have to do is examine the powers of 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 -- the columns should tend to a constant vector. 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 print(); (a) This matrix appears to stabilize to steady state given by columns of A^100, that is, approximately (0.17241,0.51724,0.31034). 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 print(); 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 print(); 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 (b) This matrix appears to flip back and forth, and does not stabilize to a steady state. However, the middle column is constant, so if the initial state of the discrete dynamical system is (0,1,0), then the dds is instantly stable. 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 print(); 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 print(); (c) This matrix appears to stabilize to steady state given by columns of A^100, which appear to be approximately (0,0,1). 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 print(); 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 print(); 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 (d) This matrix appears to stabilize to steady state given by columns of A^100, which appear to be approximately (0,0,0). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGOi8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkYvJStmb250ZmFtaWx5R0YxLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lKW1hdGhzaXplR0Y0LUYjNiVGKy1GIzYmLUYsNjlRIkFGJ0YvRjJGNUY4RjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiQ2M1EjOj1GJy8lJWZvcm1HUSZpbmZpeEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZhcC8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobWF4c2l6ZUdRKWluZmluaXR5RicvJShtaW5zaXplR1EiMUYnLyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJTBmb250X3N0eWxlX25hbWVHRlcvJSVzaXplR0Y0LyUrZm9yZWdyb3VuZEdGQy8lK2JhY2tncm91bmRHRkYtRiM2JS1GZW82M1EiW0YnL0Zpb1EncHJlZml4RicvRlxwRjpGXXAvRmBwUS50aGlubWF0aHNwYWNlRicvRmNwRmVyL0ZlcEY6RmZwRmhwRltxRl5xRmBxRmJxRmRxRmZxRmhxRmpxLUYjNiVGKy1JJ210YWJsZUdGJDYlLUkkbXRyR0YkNiUtSSRtdGRHRiQ2Iy1JI21uR0YkNjlRIjBGJ0YvRjJGNS9GOUY3RjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuL0ZpblEnbm9ybWFsRidGW29GYHMtRmFzNiMtRmRzNjlRJDAuOUYnRi9GMkY1RmdzRjtGPUY/RkFGREZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhzRltvLUZeczYlLUZhczYjLUZkczY5USQwLjVGJ0YvRjJGNUZnc0Y7Rj1GP0ZBRkRGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZoc0Zbb0Zgc0Zgcy1GXnM2JUZgc0ZhdC1GYXM2Iy1GZHM2OVEkMC4xRidGL0YyRjVGZ3NGO0Y9Rj9GQUZERkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaHNGW29GKy1GZW82M1EiXUYnL0Zpb1EocG9zdGZpeEYnRmNyRl1wRmRyL0ZjcFEydmVyeXRoaW5tYXRoc3BhY2VGJ0ZnckZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcUYrRitGKw== print(); 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 print(); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGOi8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkYvJStmb250ZmFtaWx5R0YxLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lKW1hdGhzaXplR0Y0 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 Exercise 12. (3 points for calculation, 2 for graph). Compute the total number of walks in the graph with adjacency matrix below of length at most 3. We will eyeball the resulting matrix below and add up all the entries in the result, since the (i,j)th entry gives all walks of length at most 3. To get the final answer, we'll be sneaky and multiply on left and right by a row and column of 1's. This will give us the total sum of all entries, which turns out to be 132. 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 print(); 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 print(); 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 print(); 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 For the picture of this digraph we used an external routine to create an eps file, then clicked Insert, then Image. 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