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.

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

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