A tensor is nothing but a multidimensional array. We can think of an $n-mode$ tensor as a structure whose each element has to be referred with the help of $n$ indices or $n$ axes.
Now, while reading this paper, I found a section titled as, rank $1$ tensors on page number-4.
But I seem to be a little bit confused, for I don't understand if I got that concept correctly or not. This is the precise reasoning I thought of creating the post to write out my understanding and seek validations for it's correctness.
It seems that the author says, that first we assume that say, we have $n-$ distinct axes for our reference and say, we have, an $n-mode$ tensor $X$ in $R^{I_1\times I_2\times\cdots \times I_n}.$ Furthermore, $n$ vectors $v_1,v_2,...,v_n$ are aligned in each of those $n-$ axes. We denote the $jth$ element of a vector $v_i$ as $a_i^{(j)}.$
Now, if each element of an $n-mode$ tensor, say, $x_{i_1i_2...i_n}$ is the product of the corresponding elements of those $n-$ vectors with respect to equal indices, i.e $$x_{i_1i_2...i_n}=a_{i_1}^{(1)}a_{i_2}^{(2)}\cdots a_{i_n}^{(n)}$$ then we call, $X$ as a Rank-1 tensor and use, the notation, $X=v_1\circ v_2\circ\cdots\circ v_n.$
Now, I want to know, whether I got this correctly? Any help regarding this apparent issue will be greatly appreciated.