거듭 행렬

거듭행렬(involutory matrix)은 교환 행렬의 특수한 경우이다. 반대각선이 교환행렬인 거듭행렬의 제곱 교번 특성 Jn = I 은 짝수 n 일때, 홀수 n에 대해서는 Jn = J , 여기서 n 은 임의의 정수이다. JT = J , 여기서 □T는 전치 ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 1 0 0 0 1 0 0 0 1 ) {\displaystyle {\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}\qquad } ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 1 0 0 0 1 0 0 0 1 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 0 0 1 0 1 0 1 0 0 ) {\displaystyle {\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}} ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 1 0 0 0 1 0 0 0 1 ) ( 1 0 0 0 1 0 0 0 1 ) = ( 0 0 1 0 1 0 1 0 0 ) ( 1 0 0 0 1 0 0 0 1 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 1 0 0 0 1 0 0 0 1 ) {\displaystyle {\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}} ( 1 0 0 0 1 0 0 0 1 ) ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 0 0 1 0 1 0 1 0 0 ) ( 1 0 0 0 1 0 0 0 1 ) ( 0 0 1 0 1 0 1 0 0 ) = ( 0 0 1 0 1 0 1 0 0 ) ( 0 0 1 0 1 0 1 0 0 ) ( 1 0 0 0 1 0 0 0 1 ) {\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}} 거듭행렬 간에는 교환법칙이 성립한다. 기본행렬(elementary matrix) ( a b c − a ) {\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}} a 2 + b c = 1 {\displaystyle a^{2}+bc=1} I = ( 1 0 0 0 1 0 0 0 1 ) , I − 1 = ( 1 0 0 0 1 0 0 0 1 ) {\displaystyle \mathbf {I} ={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}\;,\;\mathbf {I} ^{-1}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}} R = ( 1 0 0 0 0 1 0 1 0 ) , R − 1 = ( 1 0 0 0 0 1 0 1 0 ) {\displaystyle \mathbf {R} ={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}}\;,\;\mathbf {R} ^{-1}={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}}} S = ( + 1 0 0 0 − 1 0 0 0 − 1 ) , S − 1 = ( + 1 0 0 0 − 1 0 0 0 − 1 ) {\displaystyle \mathbf {S} ={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}}\;,\;\mathbf {S} ^{-1}={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}}} I 단위행렬 R 단위행렬에대한 순열 행렬의 특수한 경우 S 부호 행렬(signature matrix) 멱등 행렬 반대칭행렬 부호 행렬 반대각선 행렬 Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 11, 1962. Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, 2nd edition, pp 12,13 Academic Press ISBN 0-12-435560-9 매스월드