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

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

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

?

If $$ T= \left[ \begin{array}{cccc} e^{\beta J} & e^{-\beta J} \\ e^{-\beta J} & e^{\beta J} \\ \end{array} \right] $$ and $$Z = \sum_{S_i=\pm 1} ... \sum_{S_N=\pm 1} \exp{\beta J(\vec{S_1}\vec{S_2}+\vec{S_2}\vec{S_3}+...+\vec{S_{N-1}}\vec{S_N}+\vec{S_N}\vec{S_1})} $$

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

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: $$Z = \sum_{S_i=\pm 1} ... \sum_{S_N=\pm 1} \langle S_1|T|S_2\rangle\langle S_2|T|S_3\rangle...\langle S_N|T|S_1\rangle$$

?

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

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

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

?