At the moment, I try to work myself into tensors, and, especially, similarity of tensors, following the definition of matrix similarity:
Two $n\times n$ matrices $A$ and $B$ are called similar if there exists an invertible $n$-by-$n$ matrix $P$ such that $B = P^{-1}AP$.
In my situation, for example, I have a symmetric tensor $\mathcal{X} \in \mathbb{R}^{10 \times 10 \times 10}$ and a orthogonal matrix $U \in \mathbb{R}^{10 \times 10}$ and I'm looking at the following (with a $1$-mode product $\times_1$):
$$ U^{-1} \mathcal{X} U = U^T \mathcal{X} U = (\mathcal{X}^T U)^T U = (\mathcal{X} U)^T U = (\mathcal{X} \times_1 U)^T \times_1 U. $$
Does this hold? Especially, I'm wondering about equality $2$ (I'm using a rule from linear algebra) and $3$ (should hold due to the symmetry of $\mathcal{X}$).
I'm very thankful for any help!