Latex produit tensoriel

Produit tensoriel en Latex

On peut utiliser la fonction \otimes ou \bigotimes :

$$
\mathbf{v}=\left[\begin{array}{c}
v_{1} \\
v_{2} \\
\vdots \\
v_{n}
\end{array}\right], \mathbf{w}=\left[\begin{array}{c}
w_{1} \\
w_{2} \\
\vdots \\
w_{m}
\end{array}\right]
$$

Le produit tensoriel de $\mathbf{v}$ et $\mathbf{w}$ est donné par

$$
\underset{\underset{\scriptstyle\text{\bigotimes}}{\scriptstyle}}{\mathbf{v} \bigotimes \mathbf{w}} = 
\underset{\underset{\scriptstyle\text{\otimes}}{\scriptstyle}}{\mathbf{v} \otimes \mathbf{w}}=\left[\begin{array}{cccc}
v_{1} w_{1} & v_{1} w_{2} & \cdots & v_{1} w_{m} \\
v_{2} w_{1} & v_{2} w_{2} & \cdots & v_{2} w_{m} \\
\vdots & \vdots & \ddots & \vdots \\
v_{n} w_{1} & v_{n} w_{2} & \cdots & v_{n} w_{m}
\end{array}\right]
$$

$$ \mathbf{v}=\left[\begin{array}{c} v_{1} \\ v_{2} \\ \vdots \\ v_{n} \end{array}\right], \mathbf{w}=\left[\begin{array}{c} w_{1} \\ w_{2} \\ \vdots \\ w_{m} \end{array}\right] $$

Le produit tensoriel de $\mathbf{v}$ et $\mathbf{w}$ est donné par

$$ \underset{\underset{\scriptstyle\text{\bigotimes}}{\scriptstyle}}{\mathbf{v} \bigotimes \mathbf{w}} = \underset{\underset{\scriptstyle\text{\otimes}}{\scriptstyle}}{\mathbf{v} \otimes \mathbf{w}}=\left[\begin{array}{cccc} v_{1} w_{1} & v_{1} w_{2} & \cdots & v_{1} w_{m} \\ v_{2} w_{1} & v_{2} w_{2} & \cdots & v_{2} w_{m} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n} w_{1} & v_{n} w_{2} & \cdots & v_{n} w_{m} \end{array}\right] $$

Exemples

Produit tensoriel de trois vecteurs

$$ \mathbf{x}=\left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right], \mathbf{y}=\left[\begin{array}{c} y_{1} \\ y_{2} \end{array}\right], \mathbf{z}=\left[\begin{array}{c} z_{1} \\ z_{2} \end{array}\right] $$

$$
\mathbf{x} \otimes \mathbf{y} \otimes \mathbf{z}=
\begin{aligned}
\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right] \otimes\left[\begin{array}{l}
y_{1} \\
y_{2}
\end{array}\right] \otimes\left[\begin{array}{l}
z_{1} \\
z_{2}
\end{array}\right]=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right] \otimes\left[\begin{array}{l}
y_{1} z_{1} \\
y_{1} z_{2} \\
y_{2} z_{1} \\
y_{2} z_{2}
\end{array}\right] =\left[\begin{array}{l}
x_{1} y_{1} z_{1} \\
x_{1} y_{1} z_{2} \\
x_{1} y_{2} z_{1} \\
x_{1} y_{2} z_{2} \\
x_{2} y_{1} z_{1} \\
x_{2} y_{1} z_{2} \\
x_{2} y_{2} z_{1} \\
x_{2} y_{2} z_{2}
\end{array}\right]
\end{aligned}
$$

$$ \mathbf{x} \otimes \mathbf{y} \otimes \mathbf{z}= \begin{aligned} \left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \otimes\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right] \otimes\left[\begin{array}{l} z_{1} \\ z_{2} \end{array}\right]=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \otimes\left[\begin{array}{l} y_{1} z_{1} \\ y_{1} z_{2} \\ y_{2} z_{1} \\ y_{2} z_{2} \end{array}\right] =\left[\begin{array}{l} x_{1} y_{1} z_{1} \\ x_{1} y_{1} z_{2} \\ x_{1} y_{2} z_{1} \\ x_{1} y_{2} z_{2} \\ x_{2} y_{1} z_{1} \\ x_{2} y_{1} z_{2} \\ x_{2} y_{2} z_{1} \\ x_{2} y_{2} z_{2} \end{array}\right] \end{aligned} $$