ヤコビアンの計算

${\displaystyle X_1=a_{11}{Y}_1+a_{12}{Y}_2+{\mu }_1}$   ･･････(1)

${\displaystyle X_2=a_{21}{Y}_1+a_{22}{Y}_2+{\mu }_2}$   ･･････(2)

${\displaystyle \left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)+\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right)}$

${\displaystyle \left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)-\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right)=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)}$

${\displaystyle \left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)={\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)}^{-1}\left(\begin{array}{c}{X}_1-{\mu }_1\\ X_2-{\mu }_2\end{array}\right)}$

${\displaystyle \left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)=\frac{1}{a_{11}{a}_{22}-a_{12}{a}_{21}}\left(\begin{array}{cc}{a}_{22}& -a_{12}\\ -a_{21}& a_{11}\end{array}\right)\left(\begin{array}{c}{X}_1-{\mu }_1\\ X_2-{\mu }_2\end{array}\right)}$

${\displaystyle \left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)=\left(\begin{array}{c}\frac{a_{22}\left(X_1-{\mu }_1\right)-a_{12}\left(X_2-{\mu }_2\right)}{a_{11}{a}_{22}-a_{12}{a}_{21}}\\ \frac{-a_{21}\left(X_1-{\mu }_1\right)+a_{11}\left(X_2-{\mu }_2\right)}{a_{11}{a}_{22}-a_{12}{a}_{21}}\end{array}\right)}$

${\displaystyle Y_1=\frac{a_{22}\left(X_1-{\mu }_1\right)-a_{12}\left(X_2-{\mu }_2\right)}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$   ･･････(3)

${\displaystyle Y_2=\frac{-a_{21}\left(X_1-{\mu }_1\right)+a_{11}\left(X_2-{\mu }_2\right)}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$   ･･････(4)

となる．

（以降のヤコビアンの計算は，確率変数 $X_1$ ， $X_2$ ， $Y_1$ ， $Y_2$ を実現値 $x_1$ ， $x_2$ ， $y_1$ ， $y_2$ に置き換えることにする．）

よって，ヤコビアン ${\displaystyle \frac{\partial \left(y_1,y_2\right)}{\partial \left(x_1,x_2\right)}}$ は

${\displaystyle \frac{\partial \left(y_1,y_2\right)}{\partial \left(x_1,x_2\right)}=\left|\begin{array}{cc}\frac{\partial y_1}{\partial x_1}& \frac{\partial y_1}{\partial x_2}\\ \frac{\partial y_2}{\partial x_1}& \frac{\partial y_2}{\partial x_2}\end{array}\right|}$ ${\displaystyle =\left|\begin{array}{cc}\frac{a_{22}}{a_{11}{a}_{22}-a_{12}{a}_{21}}& \frac{-a_{12}}{a_{11}{a}_{22}-a_{12}{a}_{21}}\\ \frac{-a_{21}}{a_{11}{a}_{22}-a_{12}{a}_{21}}& \frac{a_{11}}{a_{11}{a}_{22}-a_{12}{a}_{21}}\end{array}\right|}$ ${\displaystyle =\frac{a_{11}{a}_{22}-a_{12}{a}_{21}}{{\left(a_{11}{a}_{22}-a_{12}{a}_{21}\right)}^2}}$ ${\displaystyle =\frac{1}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$

となる．

${\displaystyle {\displaystyle \left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)=A}}$ とおくと

${\displaystyle \left|A\right|=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|}$ ${\displaystyle =a_{11}{a}_{22}-a_{12}{a}_{21}}$

となる．したがって

${\displaystyle \frac{\partial \left(y_1,y_2\right)}{\partial \left(x_1,x_2\right)}=\frac{1}{\left|A\right|}}$

となる．

● ${\displaystyle \frac{\partial \left(y_1,y_2\right)}{\partial \left(x_1,x_2\right)}=\frac{1}{\frac{\partial \left(x_1,x_2\right)}{\partial \left(y_1,y_2\right)}}}$ の関係を用いると

${\displaystyle \frac{\partial \left(x_1,x_2\right)}{\partial \left(y_1,y_2\right)}=\left|\begin{array}{cc}\frac{\partial x_1}{\partial y_1}& \frac{\partial x_1}{\partial y_2}\\ \frac{\partial x_2}{\partial y_1}& \frac{\partial x_2}{\partial y_2}\end{array}\right|}$ ${\displaystyle =\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|}$ ${\displaystyle =\left|A\right|}$

よって

${\displaystyle \frac{\partial \left(y_1,y_2\right)}{\partial \left(x_1,x_2\right)}=\frac{1}{\frac{\partial \left(x_1,x_2\right)}{\partial \left(y_1,y_2\right)}}}$ ${\displaystyle =\frac{1}{\left|A\right|}}$

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最終更新日： 2026年7月10日