I am trying to understand this problem from the movie Good Will Hunting just because it looks fun. It asks to find the generating function for walks from points i to j. I attached a link here http://www.math.harvard.edu/archive/21b_fall_03/goodwill/ that apparently explains it. I am curious how the geometric series holds true for matrices. I am also curious how the matrix A gets thrown into the proof as well. If anyone has any information/resources on how to tackle this problem, I would be very grateful to learn.
2026-03-25 12:30:36.1774441836
Good Will Hunting Problem Reasoning
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The adjacency matrix counts the number of connections between the various nodes, in the $ij$ increasing fashion. If there is a connection, increase the corresponding count in the matrix by one. For example, in the first row in the matrix (with the nodes labeled as in the picture provided) we would have $[0,1,0,1]$, because node 1 is connected to node 2 and 4, but not connected to node 1 (itself) and node 3. The relation between the adjacency matrix and the solution is the generating function, described in your link.
The geometric series for the matrix is proved in a similar fashion to the normal geometric series. You just have to determine where the series converges. If you are interested, you can find a proof here http://www.math.uvic.ca/~dcwatson/work/geometric.pdf