鏡行列の性質３

鏡行列\(Q(\theta)\)

\begin{align}Q(\theta)=
\begin{pmatrix}
\cos 2 \theta & \sin 2 \theta \\
\sin 2 \theta & -\cos 2 \theta
\end{pmatrix}
\end{align}

と行列\(u\)

\begin{align}u=
\begin{pmatrix}
\cos \theta \\
\sin \theta
\end{pmatrix}
\end{align}

について

\begin{align}Q(\theta)u(\theta)&=
\begin{pmatrix}
\cos 2 \theta & \sin 2 \theta \\
\sin 2 \theta & -\cos 2 \theta
\end{pmatrix}
\begin{pmatrix}
\cos \theta \\
\sin \theta
\end{pmatrix}\\
&=
\begin{pmatrix}
\cos \theta \cos 2 \theta + \sin \theta \sin 2 \theta \\
\cos \theta \sin 2 \theta – \sin \theta \cos 2 \theta
\end{pmatrix}\\
&=
\begin{pmatrix}
\cos \theta \\
\sin \theta
\end{pmatrix}
\end{align}

よって

\begin{align}
Q(\theta)u(\theta)=u(\theta)
\end{align}

が成り立つ。