Thread: Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

I am seeking to understand Finite Reflection Groups and am reading Grove and Benson (G&B) : Finite Reflection Groups

Grove and Benson, in Section 21 Orthogonal Transformations in Two Dimensions define T as a linear transformation belonging to the groups of all orthogonal transformations .

G&B point out that the vector is an eigenvector having eigenvalue 1 for T and that similarly is an eigenvector with eigenvalue -1 and [see attachment]

G&B then state that if , then and T sends x to its mirror image with respect to the line L (see Figure 2.2(b) in attachement)

The transformation T is called the refection through L or the reflection along .

G&B then say "observe that for all "

Can someone help me show (explicitly and formally) that for all ?

And further (and possibly more important) can someone help me get a geometric sense of what the formula above means? ie why do G&B highlight this particular relationship?

Would very much appreciate such help

Peter

Tx = T(λ₁x₁ + λ₂x₂) = λ₁T(x₁) + λ₂T(x₂)

= λ₁x₁ - λ₂x₂ = λ₁x₁ + λ₂x₂ - 2λ₂x₂

= x - 2λ₂x₂ (this is just simple algebra, to this point).

now (x,x₂) = (λ₁x₁ + λ₂x₂,x₂) = λ₁(x₁,x₂) + λ₂(x₂,x₂)

and since x₁,x₂ are orthogonal, (x₁,x₂) = 0, therefore:

(x,x₂) = λ₂(x₂,x₂) = λ₂|x₂|².

but x₂ lies on the unit circle, therefore it has length (and thus length squared) of 1.

so (x,x₂) = λ₂, and we have:

Tx = x - 2λ₂x₂ = x - 2(x,x₂)x₂.

so why?

the quantity (x,x₂)x₂ is a little misleading (because we are dealing with UNIT vectors x₁,x₂).

normally, this is written as: [(x,x₂)/(x₂,x₂)]x₂ which is known as:

the projection of the vector x in the direction of the vector x₂. geometrically this is: "the part of x that lies on the line generated by x₂".

normally, we project a vector onto the unit vectors e₁ = (1,0) and e₂ = (0,1).

that is: if x = (x₁,x₂) = x₁e₁ + x₂e₂, then:

the projection of x in the direction of e₁ is [(x.e₁)/(e₁.e₁)]e₁

= [(x₁*1 + x₂*0)/(1*1 + 0*0)]e₁ = x₁e₁ = (x₁,0).

Thanks ... that post was most helpful, particularly the bit about the misleading nature of the formula as stated ... that clarified a few issues for me.

Geometrical interpretation is most helpful

Peter