Spin-1 matrix Sx, Sy, Sz are traceless 3*3 matrix, and have the property ##[S_i, S_j] = i\epsilon_{ijk}S_k##, and we know that ##Tr(S_i^2) = 1^2+0^2+(-1)^2=2##.(adsbygoogle = window.adsbygoogle || []).push({});

All of the above are independent of representation, of course, the trace of a matrix is representation-independent.

so, if we want to know ##Tr(S_xS_z)=?##, we can use ##Tr(S_yS_z^2)=Tr[(S_zS_y+iS_x)S_z]=Tr(S_yS_z^2)+iTr(S_xS_z)##, thus ##Tr(S_xS_z)=0##.

And my question is, what about ##Tr(S_xS_yS_xS_y)=?## Using the similar method mentioned above.

Regards!

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# Trace of the spin matrix of spin-1

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