# tensor product

## 1. some initial notes

- tensor product
- as far as I can tell, this is a generalization of the cartesian product between two vector spaces
- there is a bilinear map from the cartesian product of the vector spaces to yet another vector space

- some linguistic confusions
- the tensor product of vector spaces \(V\) and \(W\) is another vector space \(V \otimes W\)
- the tensor product of vectors \(v\) and \(w\) is an element of \(V\otimes W\), that is \(v \otimes w\)

## 2. tensor product of matrices (kronecker product)

- Linear maps can be represented by matrics
- The tensor product of these linear maps is represented by the kronecker product of their matrics
- example
- \(V\), \(W\), \(X\), \(Y\) are vector spaces
- We have linear maps \(S: V \rightarrow X\), \(T: W \rightarrow Y\)
- These are represented by \(A\) and \(B\) respectively
- Then, the linear map \(S\otimes T: V\otimes W \rightarrow X\otimes Y\) is represented by the matrix \(A \otimes B\)
- where \((A \otimes B)(v \otimes w) = (Av)\otimes (Bw)\)
- one last thing has to be said about the ordering of things in these matrices. If \(V\) and \(W\) have ordered bases \(\{v_1,...,v_m\}\) and \(\{w_1,...,w_n\}\) then the ordering of the bases of \(V\otimes W\) is \(\{v_1w_1,...,v_1w_n,v_2w_1,...,v_mw_n\}\)

- wikipedia page