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Tensor Product

Tool to perform a tensor product calculation, a kind of multiplication applicable on tensors, vectors or matrices.

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# Tensor Product

## Vector Tensor Product ⊗

### How to calculate a tensor product of matrices?

From 2 matrices $A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}$ and $B=\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}$ the tensor product noted $\otimes$ is calculated $$A \otimes B = \begin{bmatrix}a_{11}\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}&a_{12}\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix} \\ a_{21}\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix} & a_{22}\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}\end{bmatrix} = \begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&a_{12}b_{11}&a_{12}b_{12}\\a_{11}b_{21}&a_{11}b_{22}&a_{12}b_{21}&a_{12}b_{22}\\a_{21}b_{11}&a_{21}b_{12}&a_{22}b_{11}&a_{22}b_{12}\\a_{21}b_{21}&a_{21}b_{22}&a_{22}b_{21}&a_{22}b_{22}\end{bmatrix}$$

### How to calculate a tensor product of vectors?

From de 2 vectors $\vec{a} = \begin{bmatrix}a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix}$ and $\vec{b} = \begin{bmatrix}b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}$ the tensor product noted $\otimes$ is calculated $$\vec{a} \otimes \vec{b} = \vec{a} . \vec{b}^T$$ ie. like a multiplication">matrix product but with the matrix transpose of the second vector.

$$\vec{a} \otimes \vec{b} = \begin{bmatrix}a_1 b_1 & a_1 b_2 & \cdots &a_1 b_m \\ a_2 b_1 & a_2 b_2&\cdots &a_2 b_m \\ \vdots & \vdots & \ddots & \vdots \\ a_n b_1 & a_n b_2 & \cdots & a_n b_m \end{bmatrix}$$

Example: $$\begin{bmatrix} 1 \\ 2 \end{bmatrix} \otimes \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 3 & 4 \\ 6 & 8 \end{bmatrix}$$

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