= Uniaxial conductivity tensor
{title2=$\sigma=\alpha I+\gamma nn^{\mathsf T},\quad|n|=1$}
A uniaxial <electrical conductivity> has one distinguished direction $n$ and equal response in its orthogonal plane. Its eigenvalues are $\alpha$ in $n^\perp$ and $\alpha+\gamma$ along $n$, so it is invertible precisely when $\alpha(\alpha+\gamma)\ne0$. For an <electric field> perpendicular to $n$, the <electric current density> is $J=\alpha E$. Positive dissipation requires both eigenvalues to be nonnegative, a physical constraint separate from algebraic invertibility. For $D_{ij}=\epsilon_{ijk}n_k$, one has $D^2=nn^{\mathsf T}-I$, and $\sigma D^2=-\sigma$ is equivalent to $\gamma=-\alpha$; the conductivity then vanishes along $n$.
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