${{{{ \sigma} _x} ^A} _B} = {\overset{B\downarrow}{\left[\begin{matrix} 0 \\ 1\end{matrix}\right]}}$


${{{{ \sigma} _x} ^A} _B} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& 1\\ 1& 0\end{array}\right]}}$
${{{{{ \sigma} _x} ^A} ^B} = {{{\overset{A\downarrow C\rightarrow}{\left[\begin{array}{cc} 0& 1\\ 1& 0\end{array}\right]}}} {{\overset{C\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& 1\\ -{1}& 0\end{array}\right]}}}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} -{1}& 0\\ 0& 1\end{array}\right]}}$
${{{{{ \sigma} _x} _A} _B} = {{{\overset{A\downarrow C\rightarrow}{\left[\begin{array}{cc} 0& 1\\ -{1}& 0\end{array}\right]}}} {{\overset{C\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& 1\\ 1& 0\end{array}\right]}}}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} 1& 0\\ 0& -{1}\end{array}\right]}}$


${{{{ \sigma} _y} ^A} _B} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& -{i}\\ i& 0\end{array}\right]}}$
${{{{{ \sigma} _y} ^A} ^B} = {{{\overset{A\downarrow C\rightarrow}{\left[\begin{array}{cc} 0& -{i}\\ i& 0\end{array}\right]}}} {{\overset{C\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& 1\\ -{1}& 0\end{array}\right]}}}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} i& 0\\ 0& i\end{array}\right]}}$
${{{{{ \sigma} _y} _A} _B} = {{{\overset{A\downarrow C\rightarrow}{\left[\begin{array}{cc} 0& 1\\ -{1}& 0\end{array}\right]}}} {{\overset{C\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& -{i}\\ i& 0\end{array}\right]}}}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} i& 0\\ 0& i\end{array}\right]}}$


${{{{ \sigma} _z} ^A} _B} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} 1& 0\\ 0& -1\end{array}\right]}}$
${{{{{ \sigma} _z} ^A} ^B} = {{{\overset{A\downarrow C\rightarrow}{\left[\begin{array}{cc} 1& 0\\ 0& -{1}\end{array}\right]}}} {{\overset{C\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& 1\\ -{1}& 0\end{array}\right]}}}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& 1\\ 1& 0\end{array}\right]}}$
${{{{{ \sigma} _z} _A} _B} = {{{\overset{A\downarrow C\rightarrow}{\left[\begin{array}{cc} 0& 1\\ -{1}& 0\end{array}\right]}}} {{\overset{C\downarrow B\rightarrow}{\left[\begin{array}{cc} 1& 0\\ 0& -{1}\end{array}\right]}}}}} = {\overset{A\downarrow B\rightarrow}{\left[\begin{array}{cc} 0& -{1}\\ -{1}& 0\end{array}\right]}}$