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.
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$\begingroup$ What do you mean by cross products of elements of a vector? $\endgroup$– Qiaochu YuanCommented Feb 8, 2010 at 18:20
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$\begingroup$ I think we can forget the word “cross” … as far as I can tell from the wikipedia article, if you take the Kronecker product of two column vectors you get precisely a column vector as described. From an abstract perspective it is all about different concrete representations of the tensor product, which in turn is defined up to isomorphism via a universal property. The nitty-gritty details get important only in the context of computation, and there, the most important thing is just to be consistent about it. $\endgroup$– Harald Hanche-OlsenCommented Feb 8, 2010 at 18:40
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$\begingroup$ Let me elaborate on the techincal part of the comment of Harald Hanche-Olsen: The definition in Wikipedia refers to the Kronecker product of two matrices. Perhaps you should try to see what happens if you take the special case where these two matrices are column vectors $(a_i)$ and $(b_j)$. You will see that the resultant vector is $(a_i b_j)$ (in some order), which, if I understand correctly, is what you mean when you refer to "one single column vector containing the crossproduct of..." $\endgroup$– user2734Commented Feb 8, 2010 at 19:34
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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.
answered Feb 8, 2010 at 18:35
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$\begingroup$ An incorrect definition is one that departs fundamentally from standard practice. For example, $\sin(x)=x^2$ is an incorrect definition of the sine function. $\endgroup$ Commented Feb 8, 2010 at 19:35
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$\begingroup$ It is bad taste not wrong. A definition cannot be wrong. It can be useless, stupid, in conflict with accepted terminology but wrong! $\endgroup$– MBNCommented Feb 8, 2010 at 20:09
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$\begingroup$ it could be self-contradictory, or misleading (putting attention at the wrong objects, or property)... that would be a "wrong definition" ! $\endgroup$– maksCommented Mar 7, 2010 at 5:14