Normal modes of inductively coupled LC circuits (${\omega }_{\pm }$)

Two identical circuits with self-inductance $L$, capacitance $C$ and mutual inductance $\mathcal{M}$ have same-sign and opposite-sign charge modes with ${\omega }_{\pm }^2=1/[C(L\pm \mathcal{M})]$. The inductance matrix acts as the mass matrix; positivity requires $|\mathcal{M}|<L$.