行列式の定義 – TauStation

1 定義式
2 計算例
- 2.1 次数2の場合
- 2.2 次数3の場合

定義式

(1) $\begin{align*} |\boldsymbol{A}| &= \sum_{\sigma \in S_n} {\rm sgn}(\sigma) \prod_{i=1}^n a_{i\sigma(i)} \\ &= \sum_{\sigma \in S_n} {\rm sgn}(\sigma) a_{1\sigma(1)} a_{2\sigma(2)} \cdots a_{n\sigma(n)} \end{align*}$

計算例

次数2の場合

(2) $\begin{align*} |\boldsymbol{A}| &= \left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| \\ &= {\rm sgn}(\sigma_1)a_{1\sigma_1(1)}a_{2\sigma_1(2)} + {\rm sgn}(\sigma_2)a_{1\sigma_2(1)}a_{2\sigma_2(2)} \end{align*}$

ここで、

(3) $\begin{equation*} \sigma_1 = \left( \begin{array}{cc} 1 & 2 \\ 1 & 2 \end{array} \right) ,\quad \sigma_2 = \left( \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right) \end{equation*}$

(4) $\begin{equation*} {\rm sgn}(\sigma_1) = 1 ,\quad {\rm sgn}(\sigma_2) = -1 \end{equation*}$

(5) $\begin{equation*} \sigma_1(1) = 1 ,\; \sigma_1(2) = 2 ,\; \sigma_2(1) = 2 ,\; \sigma_2(2) = 1 \end{equation*}$

したがって行列式の値は、

(6) $\begin{equation*} |\boldsymbol{A}| = a_{11}a_{22} - a_{12}a_{21} \end{equation*}$

次数3の場合

(7) $\begin{align*} |\boldsymbol{A}| =& \left| \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right| \\ =& {\rm sgn}(\sigma_1)a_{1\sigma_1(1)}a_{2\sigma_1(2)}a_{3\sigma_1(3)} + {\rm sgn}(\sigma_2)a_{1\sigma_2(1)}a_{2\sigma_2(2)}a_{3\sigma_2(3)}\\ &{\rm sgn}(\sigma_3)a_{1\sigma_3(1)}a_{2\sigma_3(2)}a_{3\sigma_3(3)} + {\rm sgn}(\sigma_4)a_{1\sigma_4(1)}a_{2\sigma_4(2)}a_{4\sigma_4(3)}\\ &{\rm sgn}(\sigma_5)a_{1\sigma_5(1)}a_{2\sigma_5(2)}a_{3\sigma_5(3)} + {\rm sgn}(\sigma_6)a_{1\sigma_6(1)}a_{2\sigma_6(2)}a_{4\sigma_6(3)} \end{align*}$

ここで、

(8) $\begin{gather*} \sigma_1 = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 2 & 3 \end{array} \right) ,\quad \sigma_2 = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 & 3 & 2 \end{array} \right) \\ \sigma_3 = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 1 & 3 \end{array} \right) ,\quad \sigma_4 = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 1 \end{array} \right) \\ \sigma_5 = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 1 & 2 \end{array} \right) ,\quad \sigma_6 = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 2 & 1 \end{array} \right) \end{gather*}$

(9) $\begin{equation*} \begin{array}{ll} {\rm sgn}(\sigma_1) = \phantom{-}1 ,& {\rm sgn}(\sigma_2) = -1 \\ {\rm sgn}(\sigma_3) = -1 ,& {\rm sgn}(\sigma_4) = \phantom{-}1 \\ {\rm sgn}(\sigma_5) = \phantom{-}1 ,& {\rm sgn}(\sigma_6) = -1 \end{array} \end{equation*}$

(10) $\begin{gather*} \sigma_1(1) = 1 ,\; \sigma_1(2) = 2 ,\; \sigma_1(3) = 3 \\ \sigma_2(1) = 1 ,\; \sigma_2(2) = 3 ,\; \sigma_2(3) = 2 \\ \sigma_3(1) = 2 ,\; \sigma_3(2) = 1 ,\; \sigma_3(3) = 3 \\ \sigma_4(1) = 2 ,\; \sigma_2(2) = 3 ,\; \sigma_4(3) = 1 \\ \sigma_5(1) = 3 ,\; \sigma_5(2) = 1 ,\; \sigma_5(3) = 2 \\ \sigma_6(1) = 3 ,\; \sigma_6(2) = 2 ,\; \sigma_6(3) = 1 \end{gather*}$

したがって行列式の値は、

(11) $\begin{align*} |\boldsymbol{A}| &= a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} \\ &- a_{12}a_{21}a_{33} + a_{12}a_{23}a_{31} \\ &+ a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} \end{align*}$

定義式

計算例

次数2の場合

次数3の場合

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