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: 実装 : 2階のテンソル(テンソル) : 実装   目次

ドット積、テンソル積 $ [ X ] \cdot [ Y ] $

ドット積あるいはテンソル積 dot product, tensor product は、 算術演算として実装できる。


$\displaystyle [ X ] \cdot [ Y ] = X_{ik} Y_{kj} \{ e \} _i \otimes \{ e \} _j$     (2.72)

すなわち、


    $\displaystyle [ X ] \cdot [ Y ]$  
  $\textstyle =$ $\displaystyle \left[ \begin{array}{ccc}
Z_{00} & Z_{01} & Z_{02} \\
Z_{10} & Z_{11} & Z_{12} \\
Z_{20} & Z_{21} & Z_{22}
\end{array} \right]$  
  $\textstyle =$ $\displaystyle \left[ \begin{array}{ccc}
Z_{xx} & Z_{xy} & Z_{xz} \\
Z_{yx} & Z_{yy} & Z_{yz} \\
Z_{zx} & Z_{zy} & Z_{zz}
\end{array} \right]$ (2.73)


$\displaystyle Z_{00} = X_{00} Y_{00} + X_{01} Y_{10} + X_{02} Y_{20}$      
$\displaystyle Z_{01} = X_{00} Y_{01} + X_{01} Y_{11} + X_{02} Y_{21}$      
$\displaystyle Z_{02} = X_{00} Y_{02} + X_{01} Y_{12} + X_{02} Y_{22}$      
$\displaystyle Z_{10} = X_{10} Y_{00} + X_{11} Y_{10} + X_{12} Y_{20}$      
$\displaystyle Z_{11} = X_{10} Y_{01} + X_{11} Y_{11} + X_{12} Y_{21}$      
$\displaystyle Z_{12} = X_{10} Y_{02} + X_{11} Y_{12} + X_{12} Y_{22}$      
$\displaystyle Z_{20} = X_{20} Y_{00} + X_{21} Y_{10} + X_{22} Y_{20}$      
$\displaystyle Z_{21} = X_{20} Y_{01} + X_{21} Y_{11} + X_{22} Y_{21}$      
$\displaystyle Z_{22} = X_{20} Y_{02} + X_{21} Y_{12} + X_{22} Y_{22}$     (2.74)


$\displaystyle Z_{xx} = X_{xx} Y_{xx} + X_{xy} Y_{yx} + X_{xz} Y_{zx}$      
$\displaystyle Z_{xy} = X_{xx} Y_{xy} + X_{xy} Y_{yy} + X_{xz} Y_{zy}$      
$\displaystyle Z_{xz} = X_{xx} Y_{xz} + X_{xy} Y_{yz} + X_{xz} Y_{zz}$      
$\displaystyle Z_{yx} = X_{yx} Y_{xx} + X_{yy} Y_{yx} + X_{yz} Y_{zx}$      
$\displaystyle Z_{yy} = X_{yx} Y_{xy} + X_{yy} Y_{yy} + X_{yz} Y_{zy}$      
$\displaystyle Z_{yz} = X_{yx} Y_{xz} + X_{yy} Y_{yz} + X_{yz} Y_{zz}$      
$\displaystyle Z_{zx} = X_{zx} Y_{xx} + X_{zy} Y_{yx} + X_{zz} Y_{zx}$      
$\displaystyle Z_{zy} = X_{zx} Y_{xy} + X_{zy} Y_{yy} + X_{zz} Y_{zy}$      
$\displaystyle Z_{zz} = X_{zx} Y_{xz} + X_{zy} Y_{yz} + X_{zz} Y_{zz}$     (2.75)




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: 実装 : 2階のテンソル(テンソル) : 実装   目次
Hiroshi KAWAI 平成15年4月19日