I'm thinking in vector multiplications and thought of three natural ways to multiply them. Let me define more precisely below.
Let $a\in \mathbb{R}^{n}$ a vector with $n$ elements and $b \in \mathbb{R}^{n}$ with same dimensions.
The first method is:
$a \times b = u$, where $u \in \mathbb{R}^{n}$, and every $u_i$ element is given by $a_i \cdot b_i$.
We call it element wise multiplication in a vector. We can say
$\times:\mathbb{R}^n,\mathbb{R}^n \to \mathbb{R}^n$
The second way to do the multiplication is by inner products
$\langle a, b \rangle = \lambda$, where $\lambda \in \mathbb{R}$, a scalar given by the rule $\lambda = \sum a_i \cdot b_i$, we can also write as $a^T b$ a regular matrix multiplication operation. Doing a quick dimension analysis we have $a^T_{(1,n)} b_{(n,1)} =\lambda_{(1,1)}$, hence the scalar. Here $T$ overscripted is the transpose and the indexes are the matrix dimensions.
In this case we can say:
$\langle,\rangle:\mathbb{R}^n,\mathbb{R}^n \to \mathbb{R}$
The third and last way is doing the oposite, instead transpose the first element, we transpose the second one. That instead gives us a scalar will gives us a square matrix of size $n$, we have then
$a \otimes b = M$, where $M \in \mathbb{R}^{(n,n)}$.
Since the dimensional analysis shows $a_{(n,1)} b_{(1,n)}^T = M_{(n,n)}$.
Then we can say
$\otimes:\mathbb{R}^n,\mathbb{R}^n \to \mathbb{R}^{(n,n)}$
Here I losely used the $\otimes$ operator for lack of something better.
My question is: Did the last one has a name, studied properties and papers about it?