3. Reflections

In R$^2$, a homogeneous linear transformation $T$ is a reflection if there is some line $L$ through the origin such that for each x, $T($x$)$ is the reflection of x with $L$ as a mirror. In other words, the line segment from x to $T($x$)$ has perpendicular bisector $L$.

In R$^3$, a homogeneous linear transformation is a reflection if there is a plane $H$ through the origin such that for each x, $T($x$)$ is the reflection of x with $H$ as a mirror. In other words, the line segment from x to $T($x$)$ is perpendicular to $H$ and is bisected by $H$.

It is a fact that in R$^2$ any orthogonal matrix is either a rotation or a reflection, but as you will see from an exercise, the situation in R$^3$ is not so simple.

Suppose the homogeneous linear transformation $T$ is a reflection. Let $P$ be the matrix of $T$, and let N be any normal to the mirror, n a unit normal. Some facts:

1. The matrix $P$ of $T$ is an orthogonal matrix (because reflections are rigid).

2. $\det P = -1$ (because reflections reverse orientation).

3. $P^2 = I$ (because doing a reflection twice has no effect).

4. $T$ leaves fixed each point of its mirror.

5. $T($N$) = -$N, $T($n$) = -$n

6. $T$ has the equation $T($x$) =$   x$M$ for $M = I - 2$   n$^ t$   n$= I - \frac {\textstyle 2} {\mbox{\bf N} \cdot \mbox{\bf N}} \mbox{\bf N} ^ t \mbox{\bf N}$.

Another way to say () is that each vector in the mirror is an eigenvector of $T$ for the eigenvalue $1$. Another way to say () is that each vector perpendicular to the mirror is an eigenvector for the eigenvalue $-1$.

The fact () is easy to apply. For example, if the mirror plane is $x + 2y + 2z = 0$, then N$= (1,2,2)$ and the matrix is $I - {\frac{\textstyle 2} {\mbox{\bf N} \cdot \mbox{\bf N}}} \mbox{\bf N} ^ t \mbox{\bf N}$ $=$ $I - {\frac{2} {9}} \left[\begin{array}{c} 1\\ 2\\ 2\end{array}\right] \left[\begin{array}{rrr}1&2&2\end{array}\right]$ $=$ $\left[\begin{array}{rrr} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right] - {\frac{2} {9}} \left[\begin{array}{rrr} 1&2&2\\ 2&4&4\\ 2&4&4\end{array}\right]$ $=$ $\left[\begin{array}{rrr} \frac 79 & - \frac 49 & -\frac 49\\ - \frac 49 & \frac 19 & -\frac 89\\ - \frac 49 & -\frac 89 & \frac 19 \end{array}\right]$.

This method also works in R$^2$, where the mirror is a line with normal N. It is usually easier than the three-step method of ``rotate, easy reflection, rotate back''.

The matrices of reflections are also useful in numerical analysis, where they are called Householder transformations.

Just as for rotations, it is possible to talk about reflections that do not leave the origin fixed, i.e., whose mirrors do not go through the origin. In this case, they are not homogeneous linear transformations and cannot be described by orthogonal matrices. Also as for rotations, let's agree that we are talking about the homogeneous linear case unless it's obvious that we are not.