Some authors (v.g. the creators of Matlab, Campbell, Lo, MacKinlay (1997) in The Econometrics of Financial Markets) define the Kronecker product of two vectors as one single column vector containing the crossproduct of each lement of the first vector with each element of the second vector. This is not the usual definition in Wikipedia nor Mathworld nor other software like Mathematica. What's about that definition?, is it correct? (it seems to work in computations in Campbell, Lo, MacKinlay (1997), why does it work?, is it useful? Maybe if someone knows enough about this, he can update the Wikipedia page, otherwise I will do it with the information I receive. (This question seems not to be related at first sight with "vectorization", since vectorization of the standard definition is not equal to the nonstandard definition apparently). Many thanks in advance.
I don't understand the definition you're using, but I'll tell you that the Wikipedia definition is correct. The point of the Kronecker product is that it is a basis-dependent form of the tensor product. More precisely, given vector spaces $V$ and $W$ the vector space $V \otimes W$ is spanned by elements of the form $v \otimes w, v \in V, w \in W$. Given bases $e_1, ... e_n, f_1, ... f_m$ of $V$ and $W$, the tensor product $V \otimes W$ inherits the basis $e_i \otimes f_j, 1 \le i \le n, 1 \le j \le m$, and the Kronecker product is just what you get when you write $v \otimes w$ in this basis.