Reflection matrix

A plane reflection with unit normal $n$ is $I-2n{n}^T$; it fixes the plane $n^{\perp }$ and negates $n$.

For a nonzero normal vector $a$, reflection in $a\cdot x=0$ has matrixIts eigenvalues are $-1$ along $a$ and $1$ on $a^{\perp }$, so its determinant is $-1$.

The composition of reflections in two planes through the origin is a rotation about their line of intersection through twice the oriented angle between the planes.