A rotation along the general axis can be viewed as the composition of three consecutive rotations: (i) one that translate the general axis to the z axis; (ii) then a second rotation along the z axis, and finally; (iii) a third rotation that revert the z axis again to the original direction. Note that this composition is what we call in group theory a conjugation. Show that the conjugation does not modify the character of the transformation. The above is interpreted physically as a transformation that rotate the system by a θ angle are equivalent (belong to the same class). Hence, all matrices rotating by an angle θ, independent of their rotation axis, from a class; and different classes are labelled by different values of the rotation angle θ.

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To show that the conjugation composition of rotations does not modify the character of the transform... View the full answer