逆テンソル

もし、 $\mathrm{det} \; [ X ] \neq 0$ ならば、

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、 $\Delta_{jr}$ は

の成分 $X_{jr}$ についての余因子であり、

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

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

もし、

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)

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