{{위키데이터 속성 추적}}
'''거듭행렬'''(involutory matrix)은  [[교환 행렬]]의 특수한 경우이다.<ref>Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 11, 1962.</ref>

==교환행렬의 특수한 멱 성질==
[[반대각선]]이 교환행렬인 거듭행렬의 제곱 교번 특성

* J<sup>n</sup> = I 은 짝수 n 일때,  홀수 n에 대해서는 J<sup>n</sup> = J , 여기서 n 은 임의의 정수이다.
* J<sup>T</sup> = J    , 여기서 □<sup>T</sup>는 [[전치행렬|전치]]

:<math>
\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
</math>
:<math>
\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}
</math>

==제곱 교번 특성의 예==
:<math>
\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}
</math>
:<math>\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}
</math>
*거듭행렬 간에는 [[교환법칙]]이 성립한다.

==거듭행렬의 예==
[[기본행렬]](elementary matrix)
:<math>\begin{pmatrix}a & b \\ c & -a \end{pmatrix}</math>
:<math>a^2 + bc = 1 </math><ref>Peter Lancaster & Miron Tismenetsky (1985) ''The Theory of Matrices'', 2nd edition, pp 12,13 [[Academic Press]] {{ISBN|0-12-435560-9}}</ref>


:<math>

\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}
</math>

:<math>
\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}
</math>

:<math>
\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}
</math>
:'''I'''  [[단위행렬]]
:'''R''' [[단위행렬]]에대한 [[순열 행렬]]의 특수한 경우
:'''S''' [[부호 행렬]](signature matrix)

== 같이 보기 ==
* [[멱등 행렬]]
* [[반대칭행렬]]
* [[부호 행렬]]
* [[반대각 행렬|반대각선 행렬]]

== 각주 ==
{{각주}}
* [http://mathworld.wolfram.com/InvolutoryMatrix.html 매스월드]
[[분류:행렬]]