# Transformation properties of a tensor

The transformation properties of a tensor give

$$T^{'}_{ij} = A_{il} \ A_{jm} \ T_{lm}$$

where A describes a rotation matrix. How could I show that this is equivalent to

$$T^{'} = A \ T \ A^{T}$$

i.e. that rotation of a second-rank tensor is equivalent to a similarity transformation.

• All you need is that $A_{jm} = (A^T)_{mj}$, and then your expression is automatically the component form of the corresponding matrix multiplication. Oct 13 at 23:14