Difference Between Outer and Tensor

In summary, the conversation discusses the concept of tensor products in Hilbert spaces and their dual spaces. The outer product is defined as a linear functional acting on some elements from the two spaces, and it can be written in Dirac's Bra-Ket notation as ##\left|u\right> \left< v\right|##. It is also noted that the term "outer product" can be used in different contexts, such as in Graßmann algebra, tensor product, and cross product, but the most relevant one in this conversation is the tensor product.
  • #1
devd
47
1
Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##.
Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##.

In Dirac's Bra-Ket notation, this is written as, ##u\otimes v:=\left|u\right>\otimes \left| v\right>:=\left|u\right>\left| v\right>##What is the outer product, then?

In Dirac notation, it is written as ##\left|u\right> \left< v\right|##. Which makes it clear that the outer product is a linear functional acting on some ##v'\epsilon ~V## and giving some ##u'\epsilon~ U##, such that ##\left| u'\right>=\left<v|v'\right>\left|v\right>##.
So, would it be correct to say that (##\otimes_O## is the outer product)
$$u\otimes _Ov:=u\otimes v^* $$
for, ##u,~ v,~v^*~\epsilon ~U, V, V^*##, respectively?

Also, would it be correct to say that ##u\otimes v## is a type ##(2,0)## tensor while, ##u\otimes _O v## is a type ##(1,1)## tensor?
 
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  • #2
The term "outer product" is context sensitive. There are three multiplications which are occasionally called outer:
  1. Graßmann algebra: https://en.wikipedia.org/wiki/Exterior_algebra
  2. Tensor product: https://en.wikipedia.org/wiki/Outer_product
  3. Cross product: https://en.wikipedia.org/wiki/Cross_product
Personally I think that only the first one deserves this attribute, which makes it a tensor product with the additional requirement that ##u \wedge u = 0##. The usage you spoke about looks like you refer to the second one.
 

What is the difference between outer product and tensor product?

The main difference between outer product and tensor product lies in their definition and mathematical properties. The outer product is a binary operation that takes two vectors and produces a matrix, while the tensor product is a binary operation that takes two vectors and produces a higher-dimensional tensor.

What is the purpose of the outer product?

The outer product is useful in linear algebra for representing bilinear forms, which are functions that are linear in each of their arguments. It is also used in multivariate statistics for calculating covariance matrices.

How is the tensor product related to outer product?

The tensor product is a generalization of the outer product. In fact, the outer product can be seen as a special case of the tensor product when the tensors have a specific structure, such as being matrices.

What are the mathematical properties of the outer product and tensor product?

The outer product is commutative, meaning that the order of the operands does not affect the result. It is also distributive over addition, but not over multiplication. The tensor product, on the other hand, is neither commutative nor distributive over addition, but it is associative and has a multiplicative identity.

How are the outer product and tensor product used in physics?

In physics, the outer product is used to represent physical quantities, such as forces and electric fields, in a vector space. The tensor product is used to represent more complex physical quantities, such as stress tensors and electromagnetic fields, in higher-dimensional spaces.

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