How do you take a tensor product?

[space-time]

I have recently delved into linear algebra and multi-linear algebra. I came to learn about the concepts of linear and bi-linear maps along with bases and changes of basis, linear independence, what a subspace is and more. I then decided to move on to tensor products, when I ran into a problem:

Every video and source I come across only tells me what a tensor product is and its various properties such as:

U ⊗ V₁ + U ⊗ V₂ = U ⊗ (V₁ + V₂)

It is nice that they tell me the properties, but knowing properties such as the one above is useless if I don't actually know how to calculate what U ⊗ V is!

Can anyone please tell me how to actually calculate a tensor product (and not just the properties of tensor products)?

If you can not do this, can you at least tell me why many sources don't explain the calculation process? Is it one of those things that are generally just put into software like many partial differential equations?

 

[mathman]

I googled "tensor product". I advise you to do the same.

 

[Fredrik]

A tensor product is a bilinear map $\tau:X\times Y\to Z$ such that for each bilinear map $\sigma:X\times Y\to W$, there's a unique linear $\sigma_*:Z\to W$ such that $\sigma=\sigma_*\circ\tau$. To "calculate" $U\otimes V$ (where $U\in X$, $V\in Y$), you just plug U and V into $\tau$.
$$U\otimes V=\tau(U,V)$$ It seems that what you're concerned with is the issue of whether such a $\tau$ exists at all, and what the elements of $Z$ are like, given a specific choice of $\tau$.

There's a standard choice of $\tau$. It's discussed in this thread: https://www.physicsforums.com/threads/tensor-product-of-vector-spaces.359946/. When you read about it, you will understand why it isn't discussed much. I would say that it's discussed too much. I've seen it in physics texts, as if it's "the" definition of tensor product, but I prefer to think of the first sentence in this post as the definition, and the rest as an existence proof.

 

[atyy]

Here are some examples of tensor products given as explicit "matrix multiplication type" calculations.
http://www.quantum.umb.edu/Jacobs/QMT/QMT_AppendixA.pdf (Eq A.7-A.8)
http://www.cs.berkeley.edu/~vazirani/s09quantum/notes/lecture2.pdf (section 1.2)
http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture2.pdf (p20, Eq 19)

 

[mathwonk]

see if my notes on this page help, especially pages 28-36.

http://www.math.uga.edu/%7Eroy/845-3.pdf

 

[Stephen Tashi]

If you can not do this, can you at least tell me why many sources don't explain the calculation process?

What do you mean by "calculation". What is to be calculated? A calculation represents data and facts in some format and produces the representation of the result in some format. What formats do you wish to use for the inputs and output?

The calculation process would depend on how the mathematical objects involved are represented. For example computing the usual inner product between two vectors depends on whether the vectors are represented in cartesian coordinates or polar coordinates. There are books in applied math that teach tensor products in a concrete way by presenting everything in a Cartesian-like coordinate representation. Whether you should restort to such a book depends on your objectives. One goal of pure mathematics is to isolate important abstract concepts away from other abstract concepts - for example, to isolate the abstract properties of vectors from other concepts particular to directed line segments in Euclidean space. That leads to representing facts in a "coordinate-free" manner. People pursing applied math might find more convenient to see things done with coordinates.