Quotient theorem for Cartesian tensors
= Quotient theorem for Cartesian tensors
{title2=$T'=RTR^T$}
Suppose an array $T_{ij}$ is specified in each right-handed <orthonormal basis>. If $w_i=T_{ij}v_j$ transforms as a <vector> for every <vector> $v$, then $T$ is a second-order Cartesian <tensor> under those <basis> changes. Indeed $v'=Rv$, $w'=Rw$ imply $T'Rv=RTv$ for every $v$, hence $T'=RTR^T$. The condition on every test <vector> is essential; one contraction is insufficient.