\\ \chi \\ \omega
\end{pmatrix}
=\tilde{F}^{-1}R^T\mathbb{M}_{DV}
\begin{pmatrix}
\zeta^m \\ \chi^m \\ \omega^m
\end{pmatrix}
-\sqrt{(V^{-1})_{11}}\tilde{F}^{-1}
\begin{pmatrix}
F_{Z\eta} \\ F_{Z'\eta} \\ F_{Y\eta}
\end{pmatrix}\eta^m
\label{eq:zeta}\end{aligned}$$ We define the four-component column vector $$\begin{aligned}
\Upsilon\equiv
\begin{pmatrix}
\sqrt{(V^{-1})_{11}} \\
-\sqrt{(V^{-1})_{11}}\tilde{F}^{-1}
\begin{pmatrix}
F_{Z\eta} \\ F_{Z'\eta} \\ F_{Y\eta}
\end{pmatrix}
\end{pmatrix}
\label{eq:ups}\end{aligned}$$ and denote the first row of $R$ as $\mathbb{R}_1$ $$\mathbb{R}_1=\begin{pmatrix} R_{11} & R_{12} & R_{13} \end{pmatrix}$$ where $R_{ij}$ represents the $(i;j)$ element of $R$. We will also need the coefficient matrices $$\begin{aligned}
\mathbb{C}^{dH}=\begin{pmatrix}
C^{dH}_{Z\eta} & C^{dH}_{Z\zeta} & C^{dH}_{Z\chi} & C^{dH}_{Z\omega} \\
C^{dH}_{Z'\eta} & C^{dH}_{Z'\zeta} & C^{dH}_{Z'\chi} & C^{dH}_{Z'\omega} \\
C^{dH}_{Y\eta} & C^{dH}_{Y\zeta} & C^{dH}_{Y\chi} & C^{dH}_{Y\omega}
\end{pmatrix},\quad
\mathbb{C}^{Hd}=\begin{pmatrix}
C^{Hd}_{Z\eta} & C^{Hd}_{Z\zeta} & C^{Hd}_{Z\chi} & C^{Hd}_{Z\omega} \\
C^{Hd}_{Z'\eta} & C^{Hd}_{Z'\zeta} & C^{Hd}_{Z'\chi} & C^{Hd}_{Z'\omega} \\
C^{Hd}_{Y\eta} & C^{Hd}_{Y\zeta} & C^{Hd}_{Y\chi} & C^{Hd}_{Y\omega}
\end{pmatrix}\end{aligned}$$ Here $C^{dH}_{Z\eta}$ denotes the coefficient of $Z^\mu\eta\partial_\mu H$, while $C^{Hd}_{Z\eta}$ denotes the coefficient of $Z^\mu H\partial_\mu\eta$, and so on. If we have calculated the matrices $\mathbb{C}^{dH},\mathbb{C}^{Hd}$ and the vectors $\Upsilon$ and $\mathbb{R}_1$, then the coefficient of mass eigenstate antisymmetric $ZH\eta$ vertex ($Z^\mu(\eta\partial_\mu H-H\partial_\mu\eta)$ with all fields understood to be mass eigenstate fields) can be obtained as $$c^{as}_{ZH\eta}=\frac{\mathbb{R}_1 \mathbb{C}^{dH}\Upsilon
-\mathbb{R}_1 \mathbb{C}^{Hd}\Upsilon}{2}
\label{eq:cas}$$ while the coefficient of mass eigenstate symmetric $ZH\eta$ vertex ($Z^\mu(\eta\partial_\mu H+H\partial_\mu\eta)$ with all fields 