14
$\begingroup$

This is a beginner's question on what exactly is a tensor product, in laymen's term, for a beginner who has just learned basic group theory and basic ring theory.

I do understand from wikipedia that in some cases, the tensor product is an outer product, which takes two vectors, say $\textbf{u}$ and $\textbf{v}$, and outputs a matrix $\textbf{uv}^T$. ($\textbf{u}$ being a $m\times 1$ column vector and $\textbf{v}$ being a $n\times 1$ column vector)

How about more general cases of tensor products, e.g. in the context of quantum groups?

Sincere thanks.

  • 0
    just from having a layman's idea about what they are. $I$'m not trying to be discouraging, but rather to emphasize that you're going to have to put in a lot of wor$k$ and time for tensor products to be something you are comfortable with.2012-05-14

2 Answers 2

21

If you want to study a mathematical object, whether it is a set, manifold, group, vector space, whatever, it is often fruitful to look at natural collections of functions on that space.

Roughly, the purpose of the tensor product, $\otimes$, is to make the following statement true: $\text{functions}(X \times Y) = \text{functions}(X)\otimes \text{functions}(Y)$

The specific details about which spaces of functions to choose depend on the type of mathematical object you are interested in.

Here's a pdf that explains it better than I can, http://www.math.harvard.edu/archive/25b_spring_05/tensor.pdf

4

The difference between an ordered pair of vectors and a tensor product of two vectors is this:

If you multiply one of the vectors by a scalar and the other by the reciprocal of that scalar, you get a different ordered pair of vectors, but the same tensor product of two vectors.

Similarly with an ordered triple of vectors and a tensor product of three vectors, etc.

  • 3
    Another point on the subtlety of the phenomenon of tensors that are not pure (elementary, monomial, separable, whatever you want to call them) is that they are show up in the mathematical description of entanglement in quantum mechanics. That is, in QM the combined state space for two quantum systems is a (completed) tensor product of the original two spaces, and trying to wrap your head around entangled states of two particles is the conundrum of trying to wrap your head around the idea that some tensors are not pure.2012-05-14