If  $ 
   T=
  \left[ {\begin{array}{cccc}
   e^{\beta J} & e^{-\beta J} \\
   e^{-\beta J} & e^{\beta J} \\
  \end{array} } \right]$ 

and $Z = \sum_{S_i=\pm 1} ... \sum_{S_N=\pm 1} \exp{\beta J(\vec{S_1}\vec{S_2}+\vec{S_2}\vec{S_3}+...+\vec{N-1}\vec{N}+\vec{N}\vec{S_1})}$

Then why can we say that: $$Z = \sum_{S_i=\pm 1} ... \sum_{S_N=\pm 1} \langle S_1|T|S_2\rangle\langle S_2|T|S_3\rangle...\langle S_N|T|S_1\rangle$$

?