Linear material constitutive relations in physics can be expressed in the form
where  are two vectors representing physical quantities and  is a second-order material tensor.  In matrix form,
Examples of physical problems that fit the above template are listed in the table below.
Using  in the  matrix implies that .  Using  leads to  and .  Energy restrictions usually require  and hence we must have .  Therefore, the material properties of a transversely isotropic material are described by the matrix