I came across this quiet new to me way of defining "tensors",
That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n \times m_p} \rightarrow \mathbb{R}^{m_1 \times m_2 \times .. m_p}$ such that given matrices $V_i \in \mathbb{R}^{n \times m_i}$ for $i \in \{1,2,..,p\}$, one can write the $(i_1,i_2,..,i_p)$ component of the image $A(V_1,V_2,..,V_p)$ in terms of $np$ real numbers indexed as $A_{k_1k_2..k_p}$ with each $k_i \in \{1,2,..,n\} := [n]$ as,
$[A (V_1,V_2,..,V_p)]_{i_1i_2..i_p} = \sum_{j_1,j_2,..,j_p \in [n]}A_{j_1j_2..j_p}( V_1)_{j_1i_1} (V_2)_{j_2i_2}..(V_p)_{j_pi_p}$
- Can someone help understand if the above notion of tensors is anyhow related to the "usual" notion of a $(p,q)$ tensor of dimension $n$ being a multilinear map $\times _p V^* \times_q V \rightarrow \mathbb{R}$ where $dim(V) = n$.
It follows that the defining $np$ numbers $A_{k_1k_2..k_p}$ can be written as $A_{k_1k_2...k_p} = A(I_{n \times m_1},..,I_{n \times m_p} )_{k_1k_2...k_p} $