$2-$mode matrix tensor product

Question

Given $A$ matrix $[1\ 2;\ 3 \ 4;\ 5 \ 6]$ ($3 \times 2 $) and tensor $B$ $2 \times 2 \times 2$

I need to find a 2-mode product of $ B \times_2 A$

I have started it with computation $[B]_{(2)} , 2 \times 4$

$[B \times_2 A]_{(2)}= A [B]_{(2)}$

According to the theory I have to get the unfolding with 2 x 6, but If I take a product $ A [B]_{(2)}$ it will be $[ 3 \times 2] * [2 \times 4] = [3 \times 4]$.

What have I done wrong?

Answer 1

If $A\in \mathbb{R}^{3\times 2}$ and $\mathcal{B}\in\mathbb{R}^{2\times 2\times 2}$ then $C=\mathcal{B}\times_2 A \in \mathbb{R}^{2\times 3\times 2}$.

in kolda p.461, $\mathcal{C}=\mathcal{B}\times_2 A \Leftrightarrow C_{(2)}=AB_{(2)}$ so \begin{align*} C_{(2)}&=\left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ a_{3,1} & a_{3,2} \\ \end{array} \right)\left( \begin{array}{cc} b_{1,1,1} & b_{1,2,1} &b_{1,1,2} & b_{1,2,2} \\ b_{2,1,1} & b_{2,2,1} & b_{2,1,2} & b_{2,2,2} \\ \end{array} \right)\\ &=\left( \begin{array}{cccc} a_{1,1} b_{1,1,1}+a_{1,2} b_{2,1,1} & a_{1,1} b_{1,2,1}+a_{1,2} b_{2,2,1} & a_{1,1} b_{1,1,2}+a_{1,2} b_{2,1,2} & a_{1,1} b_{1,2,2}+a_{1,2} b_{2,2,2} \\ a_{2,1} b_{1,1,1}+a_{2,2} b_{2,1,1} & a_{2,1} b_{1,2,1}+a_{2,2} b_{2,2,1} & a_{2,1} b_{1,1,2}+a_{2,2} b_{2,1,2} & a_{2,1} b_{1,2,2}+a_{2,2} b_{2,2,2} \\ a_{3,1} b_{1,1,1}+a_{3,2} b_{2,1,1} & a_{3,1} b_{1,2,1}+a_{3,2} b_{2,2,1} & a_{3,1} b_{1,1,2}+a_{3,2} b_{2,1,2} & a_{3,1} b_{1,2,2}+a_{3,2} b_{2,2,2} \\ \end{array} \right) \end{align*}

and $$ \mathcal{C}(:,:,1)= \left( \begin{array}{ccc} a_{1,1} b_{1,1,1}+a_{1,2} b_{2,1,1} & a_{2,1} b_{1,1,1}+a_{2,2} b_{2,1,1} & a_{3,1} b_{1,1,1}+a_{3,2} b_{2,1,1} \\ a_{1,1} b_{1,2,1}+a_{1,2} b_{2,2,1} & a_{2,1} b_{1,2,1}+a_{2,2} b_{2,2,1} & a_{3,1} b_{1,2,1}+a_{3,2} b_{2,2,1} \end{array} \right) $$ $$ \mathcal{C}(:,:,2)= \left( \begin{array}{ccc} a_{1,1} b_{1,1,2}+a_{1,2} b_{2,1,2} & a_{2,1} b_{1,1,2}+a_{2,2} b_{2,1,2} & a_{3,1} b_{1,1,2}+a_{3,2} b_{2,1,2} \\ a_{1,1} b_{1,2,2}+a_{1,2} b_{2,2,2} & a_{2,1} b_{1,2,2}+a_{2,2} b_{2,2,2} & a_{3,1} b_{1,2,2}+a_{3,2} b_{2,2,2} \end{array} \right) $$