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。

$$[ X ] \cdot [ Y ] = X_{ik} Y_{kj} \{ e \} _i \otimes \{ e \} _j$$ (4.72)

すなわち、

$$[ X ] \cdot [ Y ]$$

$$= \left[ \begin{array}{ccc} Z_{00} & Z_{01} & Z_{02} \\ Z_{10} & Z_{11} & Z_{12} \\ Z_{20} & Z_{21} & Z_{22} \end{array} \right]$$

$$= \left[ \begin{array}{ccc} Z_{xx} & Z_{xy} & Z_{xz} \\ Z_{yx} & Z_{yy} & Z_{yz} \\ Z_{zx} & Z_{zy} & Z_{zz} \end{array} \right]$$ (4.73)

$$Z_{00} = X_{00} Y_{00} + X_{01} Y_{10} + X_{02} Y_{20}$$

$$Z_{01} = X_{00} Y_{01} + X_{01} Y_{11} + X_{02} Y_{21}$$

$$Z_{02} = X_{00} Y_{02} + X_{01} Y_{12} + X_{02} Y_{22}$$

$$Z_{10} = X_{10} Y_{00} + X_{11} Y_{10} + X_{12} Y_{20}$$

$$Z_{11} = X_{10} Y_{01} + X_{11} Y_{11} + X_{12} Y_{21}$$

$$Z_{12} = X_{10} Y_{02} + X_{11} Y_{12} + X_{12} Y_{22}$$

$$Z_{20} = X_{20} Y_{00} + X_{21} Y_{10} + X_{22} Y_{20}$$

$$Z_{21} = X_{20} Y_{01} + X_{21} Y_{11} + X_{22} Y_{21}$$

$$Z_{22} = X_{20} Y_{02} + X_{21} Y_{12} + X_{22} Y_{22}$$ (4.74)

$$Z_{xx} = X_{xx} Y_{xx} + X_{xy} Y_{yx} + X_{xz} Y_{zx}$$

$$Z_{xy} = X_{xx} Y_{xy} + X_{xy} Y_{yy} + X_{xz} Y_{zy}$$

$$Z_{xz} = X_{xx} Y_{xz} + X_{xy} Y_{yz} + X_{xz} Y_{zz}$$

$$Z_{yx} = X_{yx} Y_{xx} + X_{yy} Y_{yx} + X_{yz} Y_{zx}$$

$$Z_{yy} = X_{yx} Y_{xy} + X_{yy} Y_{yy} + X_{yz} Y_{zy}$$

$$Z_{yz} = X_{yx} Y_{xz} + X_{yy} Y_{yz} + X_{yz} Y_{zz}$$

$$Z_{zx} = X_{zx} Y_{xx} + X_{zy} Y_{yx} + X_{zz} Y_{zx}$$

$$Z_{zy} = X_{zx} Y_{xy} + X_{zy} Y_{yy} + X_{zz} Y_{zy}$$

$$Z_{zz} = X_{zx} Y_{xz} + X_{zy} Y_{yz} + X_{zz} Y_{zz}$$ (4.75)