 Positive inductance matrix for two identical coils

For two identical ideal coils with self-inductance $L$ and signed mutual inductance $M$, the magnetic energy isThe symmetric matrix has eigenvectors $(1,1)$ and $(1,-1)$ with eigenvalues $L+M$ and $L-M$. Strict positivity for every nonzero current vector therefore requires $L>|M|$; nonnegative energy alone permits $L=|M|$. A singular limiting matrix has a zero-energy current combination, so formulas dividing by its eigenvalues require separate treatment. Reversing one coil's winding convention changes the sign of $M$ and exchanges the two combinations.