Permutation Matrix

주어진 행렬의 행을 바꾸기 위한 행렬 (Row operation)

$$\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]=\left[\begin{array}{ll}c& d\\ a& b\end{array}\right]$$

열을 바꾸고 싶다면 permutation matrix 를 오른쪽에 놓으면 된다 (column operation).

$$\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{ll}b& a\\ d& c\end{array}\right]$$

B) 특징

1. permutation matrix 의 Inverse matrix 는 transpose 와 같다.

$$P^{-1}=P^{\top }$$

1. $n\times n$ 크기의 matrix 의 경우 총 $n!$ 개의 permutation matrix 가 존재한다.

C) Permutation Matrix

• row exchange matrix: identity matrix 에서 바꾸고 싶은 행을 바꾸면 그게 exchange matrix.
• 예시: $P_{23}=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$
  • $\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{l}1\\ 3\\ 5\end{array}\right]=\left[\begin{array}{l}1\\ 5\\ 3\end{array}\right]$
  • $\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{lll}2& 4& 1\\ 0& 0& 3\\ 0& 6& 5\end{array}\right]=\left[\begin{array}{lll}2& 4& 1\\ 0& 6& 5\\ 0& 0& 3\end{array}\right]$
• For the augmented matrix $\left[\begin{array}{ll}A& b\end{array}\right]$, that elimination step gives $\left[E_{21}A{E}_{21}b\right]$.
• $\left[\begin{array}{rll}1& 0& 0\\ -2& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{rrrr}2& 4& -2& 2\\ 4& 9& -3& 8\\ -2& -3& 7& 10\end{array}\right]=\left[\begin{array}{rrrr}2& 4& -2& 2\\ 0& 1& 1& 4\\ -2& -3& 7& 10\end{array}\right]$

D) Related

E) References