What is the intuition behind tensors? Can someone please explain the intuition behind tensors? Like an example or something of the similar kind that I should keep in mind reading the theorems about it? I can't visualize it
Thanks in advance!
 A: A possible approach to tensors, is to see them as multi-way (or multi-array). Here is a way to get an intuition of it:


*

*A set of numbers can be put together (respecting a given order) to obtain a vector (= tensor of order $1$).
$$ x_1,\ldots, x_n\in\Bbb R \qquad \to\qquad \begin{pmatrix}x_1\\\vdots \\x_n\end{pmatrix}\in\Bbb R^{n}$$

*A set of vectors (of identical dimension) can be put together (respecting a given order) to obtain a matrix (= tensor of order $2$).
$$ \begin{pmatrix}x_{1,1}\\\vdots \\x_{m,1}\end{pmatrix},\ldots,\begin{pmatrix}x_{1,n}\\\vdots \\x_{m,n}\end{pmatrix}\in\Bbb R^{n} \qquad \to\qquad\begin{pmatrix} x_{1,1}&\dots& x_{1,n}\\\vdots & & \vdots \\ x_{m,1}&\dots & x_{m,n}\end{pmatrix}\in \Bbb R^{m\times n}$$

*A set of matrices (of identical dimension) can be put together (respecting a given order) to get a tensor of order $3$.
$$\begin{pmatrix} x_{1,1,1}&\dots& x_{1,n,1}\\\vdots & & \vdots \\ x_{m,1,1}&\dots & x_{m,n,1}\end{pmatrix},\ldots,\begin{pmatrix} x_{1,1,k}&\dots& x_{1,n,k}\\\vdots & & \vdots \\ x_{m,1,k}&\dots & x_{m,n,k}\end{pmatrix}\in \Bbb R^{m\times n}\\ \qquad \to\qquad \big(x_{i_1,i_2,i_3}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k}}\in\Bbb R^{m\times n \times k}$$

*A set of tensors of order $3$ (of identical dimension) can be put together (respecting a given order) to get a tensors of order $4$.
$$\big(x_{i_1,i_2,i_3,1}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k}},\ldots,\big(x_{i_1,i_2,i_3,p}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k}}\in\Bbb R^{m\times n \times k} \qquad \to\qquad \big(x_{i_1,i_2,i_3,i_4}\big)_{\substack{1\leq i_1 \leq m\\ 1\leq i_2 \leq n\\ 1 \leq i_3 \leq k\\ 1 \leq i_4 \leq p}}\in\Bbb R^{m\times n \times k\times p}$$

*etc...


Note that I made here an example for real tensors, but, for example, you can replace $\Bbb R$ by $\Bbb C$ and the idea stays the same.
