= Orientation of an orthonormal reciprocal basis
For a real basis matrix $E$ with basis vectors as rows, the <reciprocal basis> has row matrix $E^{-T}$. If either basis is <orthonormal>, both are, and $E^{-T}=E$. Thus $E$ is an <orthogonal matrix>, with determinant either $1$ or $-1$. It is a <rotation matrix> precisely when the original ordered basis is positively oriented relative to the reference coordinates. For example, $\operatorname{diag}(1,1,-1)$ is its own orthonormal reciprocal basis matrix but is an <improper orthogonal transformation>.
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