next up previous contents
: 実装 : 2階のテンソル(テンソル) : 解説   目次

逆テンソル $ { [ X ] } ^ { -1 } $

もし、 $ \mathrm{det} \; [ X ] \neq 0$ ならば、 $ [ X ] $ は正則であり、 逆テンソル inverse tensor $ { [ X ] } ^ { -1 } $ を持つ。


$\displaystyle { [ X ] } ^ { -1 }
=
\frac{ \Delta_{jr} }{ \mathrm{det} \; [ X ] } \{ e \} _j \otimes \{ e \} _r$     (4.114)

ここで、 $\Delta_{jr}$$ [ X ] $の成分$X_{jr}$についての余因子であり、


$\displaystyle \Delta_{jr} = 1/2 e_{jkl} e_{rst} X_{ks} X_{lt}$     (4.115)

すなわち、


$\displaystyle { [ X ] } ^ { -1 }$ $\textstyle =$ $\displaystyle \frac{1}{ \mathrm{det} \; [ X ] }
\left[ \begin{array}{ccc}
Y_{00...
...} \\
Y_{10} & Y_{11} & Y_{12} \\
Y_{20} & Y_{21} & Y_{22}
\end{array} \right]$  
  $\textstyle =$ $\displaystyle \frac{1}{ \mathrm{det} \; [ X ] }
\left[ \begin{array}{ccc}
Y_{xx...
...} \\
Y_{yx} & Y_{yy} & Y_{yz} \\
Y_{zx} & Y_{zy} & Y_{zz}
\end{array} \right]$ (4.116)


$\displaystyle Y_{00} = X_{11} X_{22} - X_{12} X_{21}$      
$\displaystyle Y_{01} = X_{02} X_{21} - X_{01} X_{22}$      
$\displaystyle Y_{02} = X_{01} X_{12} - X_{02} X_{11}$      
$\displaystyle Y_{10} = X_{12} X_{20} - X_{10} X_{22}$      
$\displaystyle Y_{11} = X_{00} X_{22} - X_{02} X_{20}$      
$\displaystyle Y_{12} = X_{02} X_{10} - X_{00} X_{12}$      
$\displaystyle Y_{20} = X_{10} X_{21} - X_{11} X_{20}$      
$\displaystyle Y_{21} = X_{01} X_{20} - X_{00} X_{21}$      
$\displaystyle Y_{22} = X_{00} X_{11} - X_{01} X_{10}$     (4.117)


$\displaystyle Y_{xx} = X_{yy} X_{zz} - X_{yz} X_{zy}$      
$\displaystyle Y_{xy} = X_{xz} X_{zy} - X_{xy} X_{zz}$      
$\displaystyle Y_{xz} = X_{xy} X_{yz} - X_{xz} X_{yy}$      
$\displaystyle Y_{yx} = X_{yz} X_{zx} - X_{yx} X_{zz}$      
$\displaystyle Y_{yy} = X_{xx} X_{zz} - X_{xz} X_{zx}$      
$\displaystyle Y_{yz} = X_{xz} X_{yx} - X_{xx} X_{yz}$      
$\displaystyle Y_{zx} = X_{yx} X_{zy} - X_{yy} X_{zx}$      
$\displaystyle Y_{zy} = X_{xy} X_{zx} - X_{xx} X_{zy}$      
$\displaystyle Y_{zz} = X_{xx} X_{yy} - X_{xy} X_{yx}$     (4.118)

これは、算術演算として実装できる。

もし、 $ [ X ] $ が対称テンソルであれば、 その逆テンソル $ { [ X ] } ^ { -1 } $ も対称である。 このとき、


$\displaystyle { [ X ] } ^ { -1 }$ $\textstyle =$ $\displaystyle \frac{1}{ \mathrm{det} \; [ X ] }
\left[ \begin{array}{ccc}
Y_{00...
...} \\
Y_{01} & Y_{11} & Y_{12} \\
Y_{20} & Y_{12} & Y_{22}
\end{array} \right]$  
  $\textstyle =$ $\displaystyle \frac{1}{ \mathrm{det} \; [ X ] }
\left[ \begin{array}{ccc}
Y_{xx...
...} \\
Y_{xy} & Y_{yy} & Y_{yz} \\
Y_{zx} & Y_{yz} & Y_{zz}
\end{array} \right]$ (4.119)


$\displaystyle Y_{00} = X_{11} X_{22} - X_{12} X_{12}$      
$\displaystyle Y_{11} = X_{00} X_{22} - X_{20} X_{20}$      
$\displaystyle Y_{22} = X_{00} X_{11} - X_{01} X_{01}$      
$\displaystyle Y_{01} = X_{20} X_{12} - X_{01} X_{22}$      
$\displaystyle Y_{12} = X_{20} X_{01} - X_{00} X_{12}$      
$\displaystyle Y_{20} = X_{01} X_{12} - X_{11} X_{20}$     (4.120)


$\displaystyle Y_{xx} = X_{yy} X_{zz} - X_{yz} X_{yz}$      
$\displaystyle Y_{yy} = X_{xx} X_{zz} - X_{zx} X_{zx}$      
$\displaystyle Y_{zz} = X_{xx} X_{yy} - X_{xy} X_{xy}$      
$\displaystyle Y_{xy} = X_{zx} X_{yz} - X_{xy} X_{zz}$      
$\displaystyle Y_{yz} = X_{zx} X_{xy} - X_{xx} X_{yz}$      
$\displaystyle Y_{zx} = X_{xy} X_{yz} - X_{yy} X_{zx}$     (4.121)




next up previous contents
: 実装 : 2階のテンソル(テンソル) : 解説   目次
Hiroshi KAWAI 平成15年8月11日