# Is mirror a beam splitter with reflectence=1?

By definition from Wikipedia, the beam splitter (BS) operation $$U_{BS} = \begin{pmatrix} t & r \\ r & t \end{pmatrix} = \begin{pmatrix} \cos\theta & -i\sin\theta \\ -i\sin\theta & \cos\theta \end{pmatrix}$$, and the rotation matrix $$R=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$. Now a mirror is just a BS with $$r=1$$. Therefore it seems that $$U_{Mirror} = \begin{pmatrix} \cos\theta & 1 \\1 & \cos\theta \end{pmatrix}$$. However, it is mentioned on the same Wikipedia page that $$U_{Mirror} = R$$!

Now a mirror is just a BS with $$r=1$$. Therefore it seems that $$U_{Mirror} = \begin{pmatrix} \cos\theta & 1 \\1 & \cos\theta \end{pmatrix}$$.
This is incomplete. If $$|r|=|\sin(\theta)|=1$$, then $$\theta=\pi/2$$ and $$\cos(\theta)=0$$, so $$U_\mathrm{mirror} = \begin{pmatrix} 0 & 1 \\1 & 0 \end{pmatrix}.$$
The Wikipedia page's assertion that the unitary corresponding to a mirror is a rotation matrix $$R(\theta)$$ with variable $$\theta$$ is incorrect.