T_{1}J_{3}=\left[
\begin{array}{cc}
1 &0\\
0&-1 \\
\end{array}
\right]
\left[
\begin{array}{c}
1 \\
j \\
\end{array}
\right]
=
\left[
\begin{array}{c}
1 \\
-j \\
\end{array}
\right]
=
J_{2}
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