The polarisability tensor transforms like quadratic functions because it couples two vector quantities and thus follows second-order symmetry rules.
When a molecule is placed in an external electric field, its induced dipole moment depends not only on the strength of the field but also on its direction relative to the molecular frame. This directional dependence is captured by the polarisability tensor, whose components relate the induced dipole moment 
to the applied electric field 
. In component form, this relationship is expressed as
or, expanded for the 
-component,
When a position vector )
undergoes a symmetry operation, it transforms according to a matrix 
. If the new coordinates are 
, then:
Because and are both vectors, they must transform in the same way as the Cartesian coordinates:
is a unitary matrix and the inverse of is simply its conjugate transpose (
). Since Cartesian coordinates are real, 
and we have 
or equivalently
To find how 
transforms, we consider the relationship in the new (primed) coordinate system. Substituting eq50 into eq52, and eq53 into the resultant expression, gives:
E^{\;'}_l\;\;\;\;\;\;\;\;54)
Since the physical law must hold in the new frame, where 
,
From eq51,
\biggr\(\sum_jR_{lj}x_j\biggr\)=\sum_i\sum_j(R_{ki}x_i)(R_{lj}x_j))
Because 
, 
, 
and 
are all scalars,
\;\;\;\;\;\;\;\;56)
Comparing eq55 and eq56, the coefficients 
are identical in both expressions. This implies that under any symmetry operation, 
transforms in the same way as 
. For example, let 
and 
, so that the product 
. Applying a 
rotation about the 
-axis gives:
The product 
then transforms as:
'=(-x)(z)=-xz)
Since 
transforms with the same coefficients as the product ,
(coefficient\;for\;z)\alpha_{xz}=-\alpha_{xz})
Therefore, transforms in the same way as the quadratic function 
.