Technical Note - 行列式
行列式の定義
$n$ 次正方行列
\[A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & & a_{2n} \\ \vdots & & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{pmatrix}\]に対して、行列式（determinant） $\det A$ を下式で定義する。
\[\det A \equiv \displaystyle \sum_{\sigma \in S_n} {\rm sgn}\left( \sigma \right) \prod_{i=1}^n a_{i \sigma(i)}\]ここで、
- $S_n$：$1〜n$ の整数の並び替えの集合（全 $n!$ 通り）
- 例）$n = 3$ の場合、$S_3 = {(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1)}$
- $\sigma (i)$：並び順 $\sigma$ の $i$ 番目の要素
- 例）$n = 4, \ \sigma = (4,2,3,1)$ の場合、$\sigma (1) = 4, \ \sigma (2) = 2, \ \sigma (3) = 3, \ \sigma (4) = 1$
- ${\rm sgn}(\sigma)$：符号を表す関数で、$\sigma$ が偶置換なら1、奇置換なら-1
- 「2つを選んで置換する」という操作を、元の並び（$n=3$ なら $(1,2,3)$）に何度か適用して $\sigma$ を作る時、置換の回数が偶数なら偶置換、奇数なら奇置換
- 例）$n = 3$ の場合、
- $\sigma = (1,3,2)$：元の並び $(1,2,3)$ から1回の置換で作れるので奇置換
- $\sigma = (2,3,1)$：元の並び $(1,2,3)$ から2回の置換で作れるので偶置換
- $\sigma = (1,2,3)$：元の並び $(1,2,3)$ から0回の置換で作れるので偶置換
性質
基本変形と行列式
（1）行・列に定数（$\ne 0$）をかける
\[\det \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ {\color{red}{a_{21}}} & {\color{red}{a_{22}}} & {\color{red}{a_{23}}} & {\color{red}{a_{24}}} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} = {\color{red}{\cfrac{1}{c}}} \det \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ {\color{red}{c a_{21}}} & {\color{red}{c a_{22}}} & \color{red}{c a_{23}} & \color{red}{c a_{24}} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}\] \[\det \begin{pmatrix} a_{11} & a_{12} & {\color{red}{a_{13}}} & a_{14} \\ a_{21} & a_{22} & {\color{red}{a_{23}}} & a_{24} \\ a_{31} & a_{32} & {\color{red}{a_{33}}} & a_{34} \\ a_{41} & a_{42} & {\color{red}{a_{43}}} & a_{44} \end{pmatrix} = {\color{red}{\cfrac{1}{c}}} \det \begin{pmatrix} a_{11} & a_{12} & {\color{red}{c a_{13}}} & a_{14} \\ a_{21} & a_{22} & {\color{red}{c a_{23}}} & a_{24} \\ a_{31} & a_{32} & {\color{red}{c a_{33}}} & a_{34} \\ a_{41} & a_{42} & {\color{red}{c a_{43}}} & a_{44} \end{pmatrix}\]（2）行・列を入れ替える
\[\det \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ {\color{red}{a_{21}}} & {\color{red}{a_{22}}} & {\color{red}{a_{23}}} & {\color{red}{a_{24}}} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ {\color{blue}{a_{41}}} & {\color{blue}{a_{42}}} & {\color{blue}{a_{43}}} & {\color{blue}{a_{44}}} \end{pmatrix} = {\color{red}{-1}} \det \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ {\color{blue}{a_{41}}} & {\color{blue}{a_{42}}} & {\color{blue}{a_{43}}} & {\color{blue}{a_{44}}} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ {\color{red}{a_{21}}} & {\color{red}{a_{22}}} & {\color{red}{a_{23}}} & {\color{red}{a_{24}}} \end{pmatrix}\] \[\det \begin{pmatrix} a_{11} & a_{12} & {\color{red}{a_{13}}} & {\color{blue}{a_{14}}} \\ a_{21} & a_{22} & {\color{red}{a_{23}}} & {\color{blue}{a_{24}}} \\ a_{31} & a_{32} & {\color{red}{a_{33}}} & {\color{blue}{a_{34}}} \\ a_{41} & a_{42} & {\color{red}{a_{43}}} & {\color{blue}{a_{44}}} \end{pmatrix} = {\color{red}{-1}} \det \begin{pmatrix} a_{11} & a_{12} & {\color{blue}{a_{14}}} & {\color{red}{a_{13}}} \\ a_{21} & a_{22} & {\color{blue}{a_{24}}} & {\color{red}{a_{23}}} \\ a_{31} & a_{32} & {\color{blue}{a_{34}}} & {\color{red}{a_{33}}} \\ a_{41} & a_{42} & {\color{blue}{a_{44}}} & {\color{red}{a_{43}}} \end{pmatrix}\]（3）ある行・列に他の行・列を定数倍して加える
\[\det \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ {\color{red}{a_{21}}} & {\color{red}{a_{22}}} & {\color{red}{a_{23}}} & {\color{red}{a_{24}}} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ {\color{blue}{a_{41}}} & {\color{blue}{a_{42}}} & {\color{blue}{a_{43}}} & {\color{blue}{a_{44}}} \end{pmatrix} = \det \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ {\color{red}{a_{21}+ca_{41}}} & {\color{red}{a_{22}+ca_{42}}} & {\color{red}{a_{23}+ca_{43}}} & {\color{red}{a_{24}+ca_{44}}} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}\] \[\det \begin{pmatrix} a_{11} & a_{12} & {\color{red}{a_{13}}} & {\color{blue}{a_{14}}} \\ a_{21} & a_{22} & {\color{red}{a_{23}}} & {\color{blue}{a_{24}}} \\ a_{31} & a_{32} & {\color{red}{a_{33}}} & {\color{blue}{a_{34}}} \\ a_{41} & a_{42} & {\color{red}{a_{43}}} & {\color{blue}{a_{44}}} \end{pmatrix} = \det \begin{pmatrix} a_{11} & a_{12} & {\color{red}{a_{13}+ca_{14}}} & a_{14} \\ a_{21} & a_{22} & {\color{red}{a_{23}+ca_{24}}} & a_{24} \\ a_{31} & a_{32} & {\color{red}{a_{33}+ca_{34}}} & a_{34} \\ a_{41} & a_{42} & {\color{red}{a_{43}+ca_{44}}} & a_{44} \end{pmatrix}\]三角行列の行列式
上三角行列
\[A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1\ n-1} & a_{1n} \\ 0 & a_{22} & a_{23} & \cdots & a_{2\ n-1} & a_{2n} \\ 0 & 0 & a_{33} & \cdots & a_{3\ n-1} & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{n-1\ n-1} & a_{n-1\ n} \\ 0 & 0 & 0 & \cdots & 0 & a_{nn} \end{pmatrix}\]および下三角行列
\[A = \begin{pmatrix} a_{11} & 0 & 0 & \cdots & 0 & 0 \\ a_{21} & a_{22} & 0 & \cdots & 0 & 0 \\ a_{31} & a_{32} & a_{33} & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n-1\ 1} & a_{n-1\ 2} & a_{n-1\ 3} & \cdots & a_{n-1\ n-1} & 0 \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{n\ n-1} & a_{nn} \end{pmatrix}\]において行列式の値を計算すると、並び順 $\sigma = (1,2,3,\cdots,n)$ の項以外は積にゼロが含まれて消えるため、計算結果は対策成分の積になる：
\[\det A = \displaystyle \prod_{i=1}^n a_{ii} = a_{11}a_{22} \cdots a_{nn}\]その他の公式
\[\det A = \det A^T\] \[\det (A B) = \det A \det B\] \[\det (A^{-1}) = (\det A)^{-1}\]行列式の計算
2次元のとき
\[A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\]に対して、
\[\det A = ad - bc\]3次元のとき：サラスの方法
\[A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\]に対して、
\[\begin{pmatrix} {\color{red}{a_{11}}} & {\color{blue}{a_{12}}} & a_{13} \\ a_{21} & {\color{red}{a_{22}}} & {\color{blue}{a_{23}}} \\ {\color{blue}{a_{31}}} & a_{32} & {\color{red}{a_{33}}} \end{pmatrix}\]で同じ色がついた成分（右斜め下 に1つずつずらしていったもの）をかけ合わせて足したもの：
\[{\color{red}{a_{11}a_{22}a_{33}}} + {\color{blue}{a_{12}a_{23}a_{31}}} + a_{13}a_{21}a_{32}\]及び、
\[\begin{pmatrix} {\color{red}{a_{11}}} & {\color{blue}{a_{12}}} & a_{13} \\ {\color{blue}{a_{21}}} & a_{22} & {\color{red}{a_{23}}} \\ a_{31} & {\color{red}{a_{32}}} & {\color{blue}{a_{33}}} \end{pmatrix}\]同じ色がついた成分（左斜め下 に1つずつずらしていったもの）をかけ合わせて足したもの：
\[{\color{red}{a_{11}a_{23}a_{32}}} + {\color{blue}{a_{12}a_{21}a_{33}}} + a_{13}a_{22}a_{31}\]の差を取ることで、$A$ の行列式が計算できる：
\[\det A = {\color{red}{a_{11}a_{22}a_{33}}} + {\color{blue}{a_{12}a_{23}a_{31}}} + a_{13}a_{21}a_{32} - {\color{red}{a_{11}a_{23}a_{32}}} - {\color{blue}{a_{12}a_{21}a_{33}}} - a_{13}a_{22}a_{31}\]また、後述の余因子展開で求めることもできる。
4次元以上のとき：余因子展開
= cofactor expansion
（ToDo）