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I recently asked a question from the movie Good Will Hunting. Here is a link Good Will Hunting Problem Reasoning if you want more information. The matrix T is equal to the scalar z times the matrix shown below. I found the correct generating function for walks i to j which is S.

\begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 2 & 1\\ 0 & 2 & 0 & 0\\ 1 & 1 & 0 & 0 \end{pmatrix}

How do we know converges to zero?

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    How do you define "$T^{\infty+1}$?" – Math1000 Oct 01 '16 at 16:33
  • If you meant to ask, "for what values of $z$ does $Az$ (where $A$ is the adjacency matrix) converge," the answer is $z=0$. We instead consider the generating function $\sum_{n=0}^\infty A^nz^n$ as an element of the ring of formal power series with real coefficients, which has the unique inverse $I-Az$. – Math1000 Oct 01 '16 at 16:40

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