반대각 행렬

틀:위키데이터 속성 추적 반대각 행렬(antidiagonal matrix) 또는 역 대각 행렬은 왼쪽에서 오른쪽으로 내려가는 주대각선의 방향이 반대인 왼쪽에서 오른쪽으로 올라가는 방향의 반대각선을 갖는 행렬이다.

일반적인 주대각선과 반대각선의 행렬

반대각 행렬

$\left[\begin{array}{ccc}0& 0& a\\ 0& a& 0\\ a& 0& 0\end{array}\right]$

일반적인 주대각선의 대각행렬

$\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$

주대각선과 반대각선의 표현

$n\times nmatrix$에서,

• 주대각선의 표현

$\left(\begin{array}{ccc}{x}_{ij}& x_{ij}& x_{ij}\\ x_{ij}& x_{ij}& x_{ij}\\ x_{ij}& x_{ij}& x_{ij}\end{array}\right)$을 예약하고,

$i$는 행 그리고 $j$는 열

주 대각선 $=x_{ij}=x_{nn},i=j=n$

• 반대각선의 표현

$\left(\begin{array}{ccc}{x}_{ij}& x_{ij}& x_{ij}\\ x_{ij}& x_{ij}& x_{ij}\\ x_{ij}& x_{ij}& x_{ij}\end{array}\right)$

반대각선 $=x_{(i)((n+1)-i)}i=>1$

반대각선의 멱 성질

$\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

$\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$

$\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

• 반대각선 행렬간에는 교환법칙이 성립한다.

단위행렬

단위 행렬$I$로부터,

$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$를 예약하고,

단위 행렬의 $1$ 행과$3$행을 재배열하면,

$I^a=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

이것은 반대각행렬이면서, 순열행렬이다.

같이 보기

• 순열 행렬
• 교환 행렬

참고

• 플래닛매스