# The Unapologetic Mathematician

## Reflections

Before introducing my main question for the next series of posts, I’d like to talk a bit about reflections in a real vector space $V$ equipped with an inner product $\langle\underline{\hphantom{X}},\underline{\hphantom{X}}\rangle$. If you want a specific example you can think of the space $\mathbb{R}^n$ consisting of $n$-tuples of real numbers $v=(v^1,\dots,v^n)$. Remember that we’re writing our indices as superscripts, so we shouldn’t think of these as powers of some number $v$, but as the components of a vector. For the inner product, $\langle u,v\rangle$ you can think of the regular “dot product” $\langle u,v\rangle=u^1v^1+\dots+u^nv^n$.

Everybody with me? Good. Now that we’ve got our playing field down, we need to define a reflection. This will be an orthogonal transformation, which is just a fancy way of saying “preserves lengths and angles”. What makes it a reflection is that there’s some $n-1$-dimensional “hyperplane” $P$ that acts like a mirror. Every vector in $P$ itself is just left where it is, and a vector on the line that points perpendicularly to $P$ will be sent to its negative — “reflecting” through the “mirror” of $P$.

Any nonzero vector $\alpha$ spans a line $\mathbb{R}\alpha$, and the orthogonal complement — all the vectors perpendicular to $\alpha$ — forms an $n-1$-dimensional subspace $P_\alpha$, which we can use to make just such a reflection. We’ll write $\sigma_\alpha$ for the reflection determined in this way by $\alpha$. We can easily write down a formula for this reflection:

$\displaystyle\sigma_\alpha(\beta)=\beta-\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha$

It’s easy to check that if $\beta=c\alpha$ then $\sigma_\alpha(\beta)=-\beta$, while if $\beta$ is perpendicular to $\alpha$ — if $\langle\beta,\alpha\rangle=0$ — then $\sigma_\alpha(\beta)=\beta$, leaving the vector fixed. Thus this formula does satisfy the definition of a reflection through $P_\alpha$.

The amount that reflection moves $\beta$ in the above formula will come up a lot in the near future; enough so we’ll want to give it the notation $\beta\rtimes\alpha$. That is, we define:

$\displaystyle\beta\rtimes\alpha=\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}$

Notice that this is only linear in $\beta$, not in $\alpha$. You might also notice that this is exactly twice the length of the projection of the vector $\beta$ onto the vector $\alpha$. This notation isn’t standard, but the more common notation conflicts with other notational choices we’ve made on this weblog, so I’ve made an executive decision to try it this way.

January 18, 2010