Undergrad Is tensor product the same as dyadic product of two vectors?

[xopek]

Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two different things, such a stress tensor in different directions? Sounds too simple to be true. I tried reading about the tensor product space and it was way too abstract and every source gives a different explanation.

 

[fresh_42]

Is tensor product the same as dyadic product of two vectors?

The dyadic product is a tensor product, but a tensor product can have more than two components and is in general also the sum of e.g. dyadic products.

And dyadic multiplication is just matrix multiplication?

Yes. Column times row.

You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two different things, such a stress tensor in different directions? Sounds too simple to be true. I tried reading about the tensor product space and it was way too abstract and every source gives a different explanation.

Try this one:
https://www.physicsforums.com/insights/what-is-a-tensor/

What is a derivative? There are plenty of ways to consider it: slope, linear function, limit, etc. And like
$$
D_2(x^2)=\left.\dfrac{d}{dx}\right|_{x=2} (x^2)=4
$$
can be viewed as the linear function $x\longmapsto 4\cdot x$ can tensors be considered as multilinear functions. This is especially in physics the case. So although they are simply a number scheme, they are simultaneously functions, too.

 

[WWGD]

At the vector space level, the tensor product of two vector spaces over the same field is a new vector space
over which all bilinear maps into a fixed 3rd vector space becomes linear. It's one of those category-theoretical constructions. But maybe you were looking for more Physical intuition?

 

[xopek]

a new vector space over which all bilinear maps into a fixed 3rd vector space becomes linear.

So the tensor product creates a mathematical object called a tensor? And that object is also a bilinear map? And in case of two components it is also a 2-dimensional matrix? I know that typically a 2-dimensional matrix is a linear transformation map in a 2D vector space such as a rotation map that takes a vector and spits out another vector. But in our case this matrix (our dyadic tensor) is a bilinear map which takes two input vectors and spits out a scalar. I know that a bilinear map can become linear with one of the components held constant, i.e. what happens in a dual space V* where a covariant vector v* takes a contravariant vector v from V and produces a scalar. Is this what happens in case of the tensor product space as well?

But maybe you were looking for more Physical intuition?

Yes, that too. I was thinking of something simple such as rank-2 dyadic tensor, say in two dimensions, when two 2D vectors are "mixed" together into a 2x2 matrix aka dyadic tensor which then encodes two pieces of information coming from the two vectors, say, an angle and a direction. My main confusion is the distinction between some other object acting on a tensor, and a tensor acting on other objects. And pretty much everything about tensors.

 

[WWGD]

Yes, the tensor product $$ V \otimes W$$ of two vector spaces $$V, W $$ over the same field is a vector space whose dimension is the product of the dimensions of $$V \and W $$, so that every bilinear map $$ B: V \times W \rightarrow Z$$ factors into a linear map $$L$$ from $$ V \otimes W \rightarrow Z $$
So that the diagram commutes. Sorry, I don't know how to implement diagrams here in PF.