How is $\boldsymbol{v} \stackrel{q}{\otimes} \boldsymbol{v}$, the tensorial product of a vector by itself repeated $q$ times, is defined?

Question

The tensorial product of a vector $\boldsymbol{v}$ by itself is given as $\boldsymbol{v} \otimes \boldsymbol{v} = \boldsymbol{v} \boldsymbol{v}^\mathrm{T}$ which is a tensor of dimension 2. While i was reading a research paper (Below Eq (21) in Arxiv link), the authors defined $\boldsymbol{v} \stackrel{q}{\otimes} \boldsymbol{v}$ with $q \ge 2$. How this can actually be obtained?

Consider for instance as an example $\boldsymbol{v} = \begin{pmatrix} 1 \\ 2 \\ 3\end{pmatrix}$ and $q \in \{2, 3\}$.

Thank you

Answer 1

A couple of examples for $v=(v_1,\ldots,v_n)\in\Bbb R^n$ may help.

$$(v\otimes v)_{i,j} = v_iv_j \qquad \forall i,j =1,\ldots,n$$ $$(v\otimes v\otimes v)_{i,j,k} = v_iv_jv_k \qquad \forall i,j,k =1,\ldots,n$$ $$(v\otimes v\otimes v\otimes v)_{i,j,k,l} = v_iv_jv_kv_l \qquad \forall i,j,k,l =1,\ldots,n$$ Generally, $$(v\overset{q}\otimes v)_{i_1,\ldots,i_q} = \prod_{s=1}^qv_{i_s}\qquad \forall i_1,\ldots,i_q =1,\ldots,n$$