Eigenvectors of a rotation matrix

If a tensor is invariant under rotation about a fixed axis, the matrices representing the tensor and the rotation commute with each other. The two matrices have common eigenvectors, therefore a knowledge of eigenvectors of the rotation matrix provides us with some information about eigenvectors of the tensor. This result is applied to derive familiar representations of a transversely isotropic tensor of rank 2 and the elasticity tensor possessing tetragonal symmetry. Representation of the elasticity tensor belonging to a particular symmetry class can be achieved in an elegant manner.