= Positive inductance matrix for two identical coils
For two identical ideal coils with self-<inductance> $L$ and signed <mutual inductance> $M$, the magnetic <energy> is
$$
U=\tfrac12\begin{pmatrix}I_1&I_2\end{pmatrix}
\begin{pmatrix}L&M\\M&L\end{pmatrix}
\begin{pmatrix}I_1\\I_2\end{pmatrix}.
$$
The <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.
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