What is the name for this product?

Question

I have a vectors like:

$\vec{a} = [a_1, a_2] $

$\vec{b} = [b_1, b_2] $

And I need a vector of products of unique combinations like:

$\vec{p} = [a_1 b_1, a_1 b_2, a_2 b_1, a_2 b_2]$

does exist a name for this operation? How to describe it correctly?

Answer 1

It's close to the outer product, but because $\vec p$ isn't arranged as a $2\times 2$ matrix, I think it's more correct to say it's an instance of the Kroenecker product, which is defined for arbitrary-size matrices. In that setting it is also common to write $$ \vec p = [a_1\vec b, a_2\vec b] $$ It is a special case of a more general construction known as the tensor product.

All three of them, confusingly enough, are written as $\vec a \otimes \vec b$, except that the tensor product some times have a subscript like $\otimes _{\Bbb R}$ or $\otimes_{\Bbb C}$, depending on whether 1) there is any cause for confusion, and 2) what kind of elements the $a_i$ and $b_i$ are.