Orthogonal reflection across a hyperplane

For a unit vector $n\in\mathbb R^d$ , the orthogonal reflection across the hyperplane perpendicular to $n$ is

Sx=x-2(n\cdot x)n.

It fixes vectors perpendicular to $n$ , sends $n$ to $-n$ , and satisfies $S^2=I$ and $|Sx|=|x|$ .

Symmetry of fields

A scalar field is reflection-invariant if $f(Sx)=f(x)$ . For a vector field the natural equivariance condition is $u(Sx)=Su(x)$ , so different components can have different parities. For reflection in $z=0$ , tangential components are even in $z$ and the normal component is odd under this vector symmetry.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.