鏡行列の性質２

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\(J\)

\begin{align}J=
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\end{align}

について\(\omega = s \theta \)とすれば

\begin{align}2 \omega J Q(\theta)&=2 \omega
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
\cos 2 \theta & \sin 2 \theta \\
\sin 2 \theta & -\cos 2 \theta
\end{pmatrix}
\\
&=2 \omega
\begin{pmatrix}
-\sin 2 \theta & \cos 2 \theta \\
\cos 2 \theta & \sin 2 \theta
\end{pmatrix}\\
\end{align}

また微分すれば

\begin{align}sQ(\theta)&=
\begin{pmatrix}
\cos 2 \theta & \sin 2 \theta \\
\sin 2 \theta & -\cos 2 \theta
\end{pmatrix}\\[1.5ex]
&=2 \omega
\begin{pmatrix}
-\sin 2 \theta & \cos 2 \theta \\
\cos 2 \theta & \sin 2 \theta
\end{pmatrix}
\end{align}

が等しくなるので

\begin{align}
sQ(\theta)=2 \omega J Q(\theta)
\end{align}

が成り立つ。