The material matrix  has a symmetry with respect to a given orthogonal transformation () if it does not change when subjected to that transformation.  
For invariance of the material properties under such a transformation we require
Hence the condition for material symmetry is (using the definition of an orthogonal transformation)
Orthogonal transformations can be represented in Cartesian coordinates by a  matrix  given by
Therefore, the symmetry condition can be written in matrix form as
For a transversely isotropic material, the matrix  has the form
where the -axis is the axis of symmetry.  The material matrix remains invariant under rotation by any angle  about the -axis.