[선형대수] Lecture 18: Properties of determinants

https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/video_galleries/video-lectures/

이번 강의에서는 determination의 properties(특징)들에 대해서 배운다.
determination은 추후에 배울 eigenvalue와 매우 연관성이 높기에 잘 배워둘 필요가 있다.

먼저 다음과 같은 질문을 던진다.

왜 위 명제가 참일까? determination의 properties를 통해 증명해보자.

1. $\det(I)=1$

   • $\begin{vmatrix}1 & 0 \\ 0 & 1\end{vmatrix}=1$

2. $\text{Exchange rows : reverse sign of det.}$

   • $\begin{vmatrix}0 & 1 \\ 1 & 0\end{vmatrix}=-1$

1번과 2번 규칙을 통해서 $\det(P)= 1(even) \ or -1(odd)$라는 걸 알 수 있다.

3. (a) $\begin{vmatrix}ta & tb \\ c & d\end{vmatrix} = t\begin{vmatrix}a & b \\ c & d\end{vmatrix}$, (b) $\begin{vmatrix}a+a' & b+b' \\ c & d\end{vmatrix}=\begin{vmatrix}a & b \\ c & d\end{vmatrix}+\begin{vmatrix}a' & b' \\ c & d\end{vmatrix}$

   • (a) $\begin{vmatrix}ta & tb \\ c & d\end{vmatrix}=tad-tbc=t(ad-bc)=t\begin{vmatrix}a & b \\ c & d\end{vmatrix}$

   • (b) $\begin{vmatrix}a+a' & b+b' \\ c & d\end{vmatrix}=d(a+a')-c(b+b')=(ad-bc)+(a'd-b'c)=\begin{vmatrix}a & b \\ c & d\end{vmatrix}+\begin{vmatrix}a' & b' \\ c & d\end{vmatrix}$ (=linear each row)

4. $\text{2 equal rows} \rightarrow \det(A)=0$

   • Exchange those rows -> same matrix.

5. $\text{Subtract $l\times row \ i$ from $row \ k$ $\rightarrow$ Det. dosen't change.}$

   • $\begin{vmatrix}a & b \\ c-la & d-lb\end{vmatrix}=\begin{vmatrix}a & b \\ c & d\end{vmatrix}+\begin{vmatrix}a & b \\ -la & -lb\end{vmatrix}=\begin{vmatrix}a & b \\ c & d\end{vmatrix}$ (by 3(b))

   • $\begin{vmatrix}a & b \\ c-la & d-lb\end{vmatrix}=\begin{vmatrix}a & b \\ c & d\end{vmatrix} -l\begin{vmatrix}a & b \\ a & b\end{vmatrix}=\begin{vmatrix}a & b \\ c & d\end{vmatrix}$ (by 3(a))

6. $\text{row of zeros} \rightarrow \det(A)=0$

   • $\begin{vmatrix}0 & 0 \\ c & d\end{vmatrix}=0\begin{vmatrix}0 & 0 \\ c & d\end{vmatrix}=0$ (by 3(a))

7. $U=\begin{bmatrix}d_1 & - & . & - \\ 0 & d_2 & . & . \\ . & . & . & - \\ 0 & . & 0 & d_n \\\end{bmatrix} \rightarrow \det(U)=d_1d_2...d_n$ (product of pivots)

   • $\begin{vmatrix}d_1 & - & . & - \\ 0 & d_2 & . & . \\ . & . & . & - \\ 0 & . & 0 & d_n \\\end{vmatrix} \rightarrow \begin{bmatrix}d_1 & 0 & . & 0 \\ 0 & d_2 & . & . \\ . & . & . & 0 \\ 0 & . & 0 & d_n \\\end{bmatrix}\rightarrow d_1\begin{bmatrix}1 & 0 & . & 0 \\ 0 & d_2 & . & . \\ . & . & . & 0 \\ 0 & . & 0 & d_n \\\end{bmatrix} \rightarrow d_2d_1\begin{bmatrix}1 & 0 & . & 0 \\ 0 & 1 & . & . \\ . & . & . & 0 \\ 0 & . & 0 & d_n \\\end{bmatrix} \rightarrow ... \rightarrow d_n...d_2d_1\det(I)=d_n...d_2d_1$

8. (a) $\text{$A$ is singular} \rightarrow \det(A)=0$, (b) $\text{$A$ is invertible} \rightarrow \det(A)\ne0$

   • (a) $\text{$A$ is singular} \rightarrow \text{$A$ has row of zeros} \rightarrow \det(A)=0$

   • (b) $\text{$A$ is invertible} \rightarrow A\rightarrow U \rightarrow D \rightarrow d_1d_2...d_n$

9. $\det(AB)=\det(A)\det(B)$

   • $\det(A^{-1})=\frac{1}{\det(A)}$ (여기서 $\det(A)\ne0$ 라는 걸 알 수 있다.)

     • $\det(A^{-1}A)=\det(I)=1=\det(A^{-1})\det(A)$
       $\therefore \det(A^{-1})=\frac{1}{\det(A)}$

   • $\det(A^2)=\det(A)^2$

   • $\det(2A)=2^n\det(A)$

     • $A:n\times n$
       $\det(2A)=\begin{vmatrix}2 & 0 & ... & 0 \\ 0 & 2 & ... & 0 \\ 0 & 0 & ... & 0 \\ . & . & ... & . \\ 0 & 0 & ... & 2 \\ \end{vmatrix}\det(A)=2^n\det(A)$

10. $\det(A^T)=\det(A)$

    • $\begin{vmatrix}a & b \\ c & d\end{vmatrix}=\begin{vmatrix}a & c \\ b & d\end{vmatrix}$

    • $\det(LU)=\det(U^TL^T) \\ \det(L)\det(U)=\det(U^T)\det(L^T)$
      $L$, $U$는 7번 특징에 의해서 대각성분들이 값만 곱해줌으로써 det.가 가능하다. 따라서 Transpose가 진행되도 값을 변하지 않는다.

    • 또한 이 특징에 따라 위 1~9번 특징들이 모두 row뿐만 아니라 column에도 똑같이 적용될 수 있다는 것을 의미한다. (ex. column change, column of zeros, ...)