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I'm looking for something similar to Kronecker's product that creates matrix based on two vectors I'm thinking about either:

$$ (1,2,3) @ (4,5,6) = \begin{bmatrix} (1, 4) & (2, 4) & (3, 4) \\[0.3em] (1, 5) & (2, 5) & (3, 5) \\[0.3em] (1, 6) & (2, 6) & (3, 6) \end{bmatrix} $$

or

$$ (1,2,3) @ (4,5,6) = \begin{bmatrix} 1 \cdot 4 & 2 \cdot 4 & 3 \cdot 4 \\[0.3em] 1 \cdot 5 & 2 \cdot 5 & 3 \cdot 5 \\[0.3em] 1 \cdot 6 & 2 \cdot 6 & 3 \cdot 6 \end{bmatrix} $$

where obviously @ is my imaginary operation.

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If they were sets, though, then Cartesian products. Hmm I have no other idea. –  FrenzY DT. Aug 22 '12 at 1:24
    
$v\cdot w^t$?${}$ –  tomasz Aug 22 '12 at 1:25
    
@tomasz for second one you're obviously right, I don't know how I missed that. –  Marcin Raczkowski Aug 22 '12 at 1:32

2 Answers 2

up vote 1 down vote accepted

The second one is sometimes called the outer product.

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I don't know why this question has a negative vote. If you have row vectors $u = (1,2,3)$ and $v = (4,5,6)$, the latter matrix is $v^Tu$. The former isn't really a matrix in the sense I know of, because its entries aren't numbers.

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Well, the former is a matrix with entries in the ring $R^2$, where $R$ is the base ring of the vectors. I doubt there's any common way to denote such an operation, though. Doesn't seem very natural. –  tomasz Aug 22 '12 at 1:27
    
IIRC matrices are more versatile, and as long as entries belong to a ring that has some operations defined, they are valid. –  Marcin Raczkowski Aug 22 '12 at 1:29
    
@MarcinRaczkowski: Yes. I'm just saying that the operation $(R^{n\times 1})^2\to (R^2)^{n\times n}$ doesn't look natural. –  tomasz Aug 22 '12 at 1:31

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