I understand how to do a tensor product of 2 vectors, but I am not quite sure how to incorporate a third one. Can someone explain to me how to compute tensor product if I have 3 vectors
$$u\otimes v\otimes w$$ $$u=(1,1)$$ $$v=(1,-2)$$ $$w=(-1,3)$$
I can't find any examples online, and I feel like I will understand it most if I see an example.
Thank you.
1 Answer
Well, it's easier if you write them in index notation
$u^{i} = \begin{bmatrix}1\\1\end{bmatrix} $ $v^{i} = \begin{bmatrix}1\\-2\end{bmatrix} $ $w^{i} = \begin{bmatrix}-1\\3\end{bmatrix} $
then their tensor product $(u\otimes v\otimes w)$ simply has components: $$(u\otimes v\otimes w)^{ijk} = u^{i}v^{j}w^{k}$$
To be completely explicit:
$$\begin{array} (u\otimes v\otimes w)^{111} = u^{1}v^{1}w^{1} &= (1)(1)(-1) &= -1\\ (u\otimes v\otimes w)^{112} = u^{1}v^{1}w^{2} &= (1)(1)(3) &= 3\\ (u\otimes v\otimes w)^{121} = u^{1}v^{2}w^{1} &= (1)(-2)(-1) &= 2\\ (u\otimes v\otimes w)^{122} = u^{1}v^{2}w^{2} &= (1)(-2)(3) &= -6\\ (u\otimes v\otimes w)^{211} = u^{2}v^{1}w^{1} &= (1)(1)(-1) &= -1\\ (u\otimes v\otimes w)^{212} = u^{2}v^{1}w^{2} &= (1)(1)(3) &= 3\\ (u\otimes v\otimes w)^{221} = u^{2}v^{2}w^{1} &= (1)(-2)(-1) &= 2\\ (u\otimes v\otimes w)^{222} = u^{2}v^{2}w^{2} &= (1)(-2)(3) &= -6\\ \end{array}$$
So you see, in order to write the components of this tensor as a matrix you would need to arrange them in a cube. But the components are there.
