Currently I am starting to study tensor calculus and I came across the definition of the tensor product, and more specifically the definition of tensor rank (ex. a tensor product of 2 rank 1 tensors (vectors) give me a rank 2 tensor, that is a matrix). However I have previously worked with block matrices and from the definition given in https://en.wikipedia.org/wiki/Kronecker_product I was wondering if when someone talks about a tensor rank 3 or 4 we can represent that 3D/4D matrix just as a simple block matrix? maybe I am being sloppy with the definitions but is just to maybe have an intuition of what is going on.
Also my second question is if given 2 vectors
\begin{aligned} \displaystyle \mathbf {v} ={\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}},\ \mathbf {w} ={\begin{bmatrix}w_{1}\\w_{2}\ \end{bmatrix}} \end{aligned} I perform the tensor product $\mathbf {v} \otimes \mathbf {w}$ why do I obtain the following matrix (as explained in https://en.wikipedia.org/wiki/Tensor_product):
\begin{aligned} \begin{bmatrix}v_{1}w_{1}&v_{1}w_{2}\\v_{2}w_{1}&v_{2}w_{2}\\ \end{bmatrix} \end{aligned}
Instead of \begin{aligned} {\begin{bmatrix}v_{1}w_{1}\\v_{1}w_{2} \\v_{2}w_{1} \\ v_{2}w_{2} \\ \end{bmatrix}}? \end{aligned}
Maybe it is a bit related to the first part of the tensor product (from my understanding both the matrix and the 4 elements vector are tensors of rank 2 since I need 2 indices to specify an element, and the vector could be seen as a block matrix/vector)
If someone could help me understand or correct me if my understanding is faulty I will appreciate it