The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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January 18, 2010 - Posted by | Algebra, Geometry, Linear Algebra

6 Comments »

  1. I am still with you, even though I’ve mostly been lurking uncommented under the distractions of flu, job search, and my son being home some weekends from Law School, and sleeping all day, blocking my access to his PC.

    You are making wonderful presentations each and every time!

    Comment by Jonathan Vos Post | January 18, 2010 | Reply

  2. For a moment, I thought this post would be *reflective*, full of emotions and things like that.

    Comment by christopherdrup | January 18, 2010 | Reply

  3. [...] Lemma on Reflections Here’s a fact we’ll find useful soon enough as we talk about reflections. Hopefully it will also help get back into thinking about linear transformations and inner product [...]

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  4. [...] are basically collections of vectors in some inner product space , but with each vector comes a reflection and we want these reflections to play nicely with the vectors themselves. In a way, each point acts [...]

    Pingback by Root Systems « The Unapologetic Mathematician | January 20, 2010 | Reply

  5. [...] Let’s take a root system in the inner product space . Each vector in gives rise to a reflection in , the group of transformations preserving the inner product on . So what sorts of [...]

    Pingback by The Weyl Group of a Root System « The Unapologetic Mathematician | January 21, 2010 | Reply

  6. [...] laying down some definitions on reflections, we defined a root system as a collection of vectors with certain properties. Specifically, each [...]

    Pingback by Root Systems Recap « The Unapologetic Mathematician | March 12, 2010 | Reply


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