diagonal with entries $\left(C_{1},\ldots,C_{N}\right)$, $C_{i}$ being the total capacitance shunting the junction $i$. This is a valid physical assumption since it corresponds to having no direct electrostatic dipole-dipole interaction between junction terminals. Such an assumption will keep our discussion simple although the case of non-diagonal $\mathbf{C}_{0}$ will not change any of the results.

$\mathbf{R}$ is a $M\times N$ matrix generating the couplings between qubits and internal modes. $\mathbf{R}$ consists of row vectors $r_{k}=\left(r_{k1}\ldots r_{kN}\right)$ with $r_{k}^{T}r_{k}=\mathbf{A}_{k}$. $\mathbf{M}_{0}$ matrix is diagonal with entries $\left(1/L_{1},\ldots,1/L_{N},1/L_{R_{1}},\ldots,1/L_{R_{M}}\right)$ where $L_{R_{k}}=1/\omega_{R_{k}}^{2}$ for $1\leq k\leq M$. Here we replaced the Josephson junction $i$ with the qubit inductance $L_{i}$ such that $1/\sqrt{L_{i}C_{i}}=\omega_{i}$.

We do a capacitance rescaling [@Brito] $\mathbf{\Phi}_{J}\rightarrow\mathbf{C}_{0}^{1/2}\mathbf{\Phi}_{J}$ to transform the capacitance matrix $\mathbf{C}$ as follows

$$\mathbf{C}\rightarrow\left(\begin{array}{cc}
\mathbf{1}_{N\times N} & -\mathbf{C}_{0}^{1/2}\mathbf{R}^{T}\\
-\mathbf{R}\mathbf{C}_{0}^{1/2} & \mathbf{1}_{M\times M}+\mathbf{R}\mathbf{C}_{0}\mathbf{R}^{T}
\end{array}\right)$$

and $\mathbf{M}_{0}$ transforming into the diagonal matrix with entries $(\omega_{1}^{2},\ldots,\omega_{N}^{2},\omega_{R_{1}}^{2},\ldots,\omega_{R_{M}}^{2})$. At this point

We then apply the transformation

$$\mathbf{T}=\left(\begin{array}{cc}
\mathbf{1}_{N\times N} & \mathbf{C}_{0}^{1/2}\mathbf{R}^{T}\\
\mathbf{0}_{M\times N} & \mathbf{1}_{M\times M}
\end{array}\right)\label{eq:T-transformation}$$

to reduce the capacitance matrix to identity

$$\mathbf{C}\rightarrow\mathbf{T}^{T}\mathbf{C}\mathbf{T}=\mathbf{1}$$

Then $\mathbf{M}_{0}$ transforms to $\mathbf{M}_{1}$ as

$$\begin{aligned}
\mathbf{M}_{1} & = & \mathbf{T}^{T}\mathbf{C}_{0}^{-1/2}\mathbf{M}_{0}\mathbf{C}_{0}^{-1/2}\mathbf{T}\nonumber \\
 & = & \left(\begin{array}{cc}
\mathbf{\Omega}_{J}^{2} & \mathbf{\Omega}_{J}^{2}\mathbf{C}_{0}^{1/2}\mathbf{R}^{T}\\
\mathbf{R}\mathbf{C}_{0}^{1/2}\mathbf{\Omega}_{J}^{2} & \mathbf{\Omega}_{R'}^{2}
\end{array}\right)\label{eq:M1-matrix}\end{aligned}$$

where $\mathbf{\Omega}_{R'}^{2}=\mathbf{\Omega}_{R}^{2}+\mathbf{R}\mathbf{C}_{0}^{1/2}\mathbf{\Omega}_{J}^{2}\mathbf{C}_{0}^{1/2}\mathbf{R}^{T}$, $\mathbf{\Omega}_{J}$ is diagonal with entries $\left(\omega_{1},\ldots,\omega_{N}\right)$, $\omega_{i}=1/\sqrt{L_{i}C_{i}}$ for 