\hat{R}(\xi +\pi/2) \hat{\Theta}^{-1}\hat{R}^{-1}(\xi+\pi/2).$$ At this point, it is convenient to introduce the bra-ket notation for the row and column vector of $\bm{\omega}_{\perp}$ as $\bra{\omega_{\perp}}$ and $\ket{\omega_{\perp}}$, respectively. With this notation, Eq. (\[eq:lin-omg2\]) can be put in the form of $$\begin{aligned}
    \ket{\ddot{\tilde{\omega}}_{\perp}}= -\hat{H}\ket{\tilde{\omega}_{\perp}},
\label{eq:lin-omg3}\end{aligned}$$ with $$\begin{aligned}
   \ket{\tilde{\omega}_{\perp}} \equiv \hat{J}^{1/2}\ket{\omega_{\perp}},
   \quad \hat{H} \equiv \frac{g}{a}\hat{J}^{-1/2}(\hat{\Gamma} -1 )\hat{J}^{-1/2},
\label{eq:lin-omg4}\end{aligned}$$ where $\hat{H}$ is symmetric. The eigenvalue equation $$\begin{aligned}
 \hat{H} \ket{\tilde{\omega}_{j}}= \omega_{j}^2 \ket{\tilde{\omega}_{j}}
\label{eq:def-omgpr}\end{aligned}$$ determines the two oscillation modes with $j=p$ or $r$, whose frequencies are given by $$\omega_{p,r}^2 = \frac{1}{2}\left[(H_{11}+H_{22})
      \pm \sqrt{(H_{11}-H_{22})^2 + 4H_{12}^2}\right]
\label{eq:def-omega_pr}$$ with $$\omega_{p} \ge \omega_{r}.
\label{ineq:omega_pr}$$ Here, $H_{ij}$ denotes the $ij$ component of $\hat H$. The orthogonal condition for the eigenvectors $\ket{\tilde{\omega}_{p}}$ and $\ket{\tilde{\omega}_{r}}$ can be written using $\hat{\varepsilon}$ as $$\begin{aligned}
 \ket{\tilde{\omega}_{p}} &= \hat{\varepsilon} \ket{\tilde{\omega}_{r}},\quad
 \ket{\tilde{\omega}_{r}} = -\hat{\varepsilon} \ket{\tilde{\omega}_{p}},  \\
 \bra{\tilde{\omega}_{r}} &=  \bra{\tilde{\omega}_{p}}\hat{\varepsilon}, \quad
 \bra{\tilde{\omega}_{p}} =  -\bra{\tilde{\omega}_{r}}\hat{\varepsilon}. \end{aligned}$$ In the case of zero skew angle, $\xi=0$, we have $$\begin{aligned}
\omega_p^2 &=\left({g\over a}\right){1/\phi-1\over\alpha}\equiv \omega_{p0}^2,
\label{def:omega_p0}
\\
\omega_r^2 &=\left({g\over a}\right){1/\theta-1\over\beta}\equiv \omega_{r0}^2,
\label{def:omega_r0}\end{aligned}$$ and the eigenvectors $\ket{\omega_{p}}$ and $\ket{\omega_{r}}$ are parallel to the $x$ and the $y$ axes, thus these modes correspond to the pitching and the rolling oscillations, respectively. This correspondence holds for $|\xi|\ll 1$ and $\omega_{p0}>\omega_{r0}$ as for a typical rattleback parameter, 