{#s3}
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We illustrate the above results concerning $\mathcal{PT}$-symmetric quantum mechanics using the finite-dimensional $2\times2$ symmetric matrix Hamiltonian $$H=\left(\begin{array}{cc} re^{i\theta} & s \cr s & re^{-i\theta}
\end{array}\right),
\label{e21}$$ where the three parameters $r$, $s$, and $\theta$ are real. This Hamiltonian is not Hermitian in the usual Dirac sense, but it is $\mathcal{PT}$ symmetric, where the parity operator is [@rr13] $$\mathcal{P}=\left(\begin{array}{cc} 0 & 1 \cr 1 & 0
\end{array}\right)
\label{e20}$$ and $\mathcal{T}$ performs complex conjugation. (Note that $\mathcal{T}$ does [*not*]{} perform Hermitian conjugation, or else it would not leave the commutation relation $[x,p]=i$ invariant.)

There are two parametric regions for this Hamiltonian. When $s^2<r^2\sin^2
\theta$, the energy eigenvalues form a complex conjugate pair. This is the region of broken $\mathcal{PT}$ symmetry. On the other hand, if $s^2\geq r^2
\sin^2\theta$, then the eigenvalues $\varepsilon_\pm=r\cos\theta\pm\sqrt{s^2-r^2
\sin^2\theta}$ are real. This is the region of unbroken $\mathcal{PT}$ symmetry. In the unbroken region the simultaneous eigenstates of the operators $H$ and $\mathcal{PT}$ are $$|\varepsilon_+\rangle=\frac{1}{\sqrt{2\cos\alpha}}
\left(\begin{array}{c}e^{i\alpha/2}\cr e^{-i\alpha/2}\end{array}\right)
\quad{\rm and}\quad|\varepsilon_-\rangle=\frac{i}{\sqrt{2\cos\alpha}}
\left(\begin{array}{c}e^{-i\alpha/2}\cr-e^{i\alpha/2}\end{array}\right),
\label{e22}$$ where we set $\sin\alpha =(r/s)\,\sin\theta$. It is easily verified that $(\varepsilon_{\pm},\varepsilon_{\pm})=\pm1$ and that $(\varepsilon_{\pm},
\varepsilon_{\mp})=0$, recalling that $(u,v)=(\mathcal{PT}u)\cdot v$. Therefore, with respect to the $\mathcal{PT}$ inner product, the resulting vector space spanned by energy eigenstates has 