What is the Standard Definition of a Tensor Product of Two Vectors?

[Bashyboy]

I am have been searching for the of a tensor product of two vectors, but found seemingly conflicting definitions. For example, one source definition was, roughly, that the tensor product of two vectors was another column vector in a higher dimensional space, and another defined the tensor product of two vectors as resulting in a matrix.

So, which of the two is correct?

 

[WWGD]

Maybe the issue is one of dualization, but the product of a k-tensor T and an n-tensor S is the (k+n)-tensor given by ##T\otimes S:= T(x_1,..,x_k) \otimes S(x_{k+1},..,x_{k+n})## EDIT: Here I assume vectors are treated as 0-tensors.

 

[micromass]

I am have been searching for the of a tensor product of two vectors, but found seemingly conflicting definitions. For example, one source definition was, roughly, that the tensor product of two vectors was another column vector in a higher dimensional space, and another defined the tensor product of two vectors as resulting in a matrix.

So, which of the two is correct?

You will need to show those sources.

 

[HallsofIvy]

I am have been searching for the of a tensor product of two vectors, but found seemingly conflicting definitions. For example, one source definition was, roughly, that the tensor product of two vectors was another column vector in a higher dimensional space,

I have never heard of this. What was the source? Perhaps you are confusing a tensor product with a direct product.

and another defined the tensor product of two vectors as resulting in a matrix.

I doubt you read this correctly. The tensor product of two vectors (i.e. two first order tensors) is a second order tensor tensor which in a given coordinate system can be represented by a matrix. You should be careful to distinguish between those two concepts. Physically, velocity is a vector. But a velocity vector with speed v can be represented by the array (v, 0, 0) in one coordinate system, (0, v, 0) in another, and generally (a, b, c) with [itex]a^2+ b^2+ c^2= v^2[/itex]. A tensor can be represented as an array of numbers in a specific coordinate system.

So, which of the two is correct?

Strictly speaking, neither is correct! But the second is a little closer.

 

[WWGD]

Maybe if you had a natural isomorphism ##V \rightarrow V^{*} ## (e.g., having an associated non-degenerate form ), you can see your vectors naturally/canonically as linear maps and then you get the standard definition ## v\otimes w \approx. (T\otimes S)(x,y):=T(x)S(y)##.