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I've been thinking about different products of vectors and the relationships between them, but am bumping hard into the limits of my knowledge. I was wondering if anyone has any thoughts.

Given two (Euclidean) vectors a and b, we can define a scalar product a.b which gives a scalar of magnitude |a||b|cos t where t is angle between them.

We can also define a tensor product ab whereby we get a matrix whose (ij)th element is the product of the ith element of a and the jth element of b, i.e. (ab)_ij =a_i b_j.

We know that if a and b are perpendicular then a.b=0 and also that Tr(ab)=0.

It seems ike there should be more statement like this which connect the two products, but I am a bit stuck as to what they might be. Does anyone have any thoughts or could anyone point me at some further reading on this?

Many thanks.

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    $\begingroup$ In general, $\mathbf{a} \cdot \mathbf{b} = \operatorname{Tr}(\mathbf{a}\mathbf{b})$. I'm not sure there's more to say. $\endgroup$
    – Matthew Leingang
    Sep 6, 2016 at 15:27

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In fact, if $\mathbf a\otimes\mathbf b$ denotes the tensor product, i.e. $$ (\mathbf a\otimes\mathbf b)_{ij}:=a_i b_j, $$ then $$ \mathbf a\cdot\mathbf b=\sum_{i=1}^na_ib_i=\sum_{i=1}^n(\mathbf a\otimes\mathbf b)_{ii}=\mathrm{Tr} (\mathbf a\otimes\mathbf b) $$

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