# Thread: Eigenvalues and Eigenvectors in the Context of Finite Reflection Groups

1. ## Eigenvalues and Eigenvectors in the Context of Finite Reflection Groups

Grove and Benson: FInite Reflection Groups in the context of developing the basic theory of finite reflection groups give the following problem in Chapter 2

Exercise 2.1

Verify that

$x_1 = ( cos \ \theta / 2 , sin \ \theta /2).$

$x_2 = ( -sin \ \theta / 2 , cos \ \theta /2).$

are eigenvectors with respective eigenvalues 1 and -1 for the matrix

$B = \left(\begin{array}{cc}{cos \ \theta}&{sine \ \theta}\\{sin \ \theta}&{- cos \ \theta}\end{array}\right )$

I would appreciate help with this problem.

Peter

2. ## Re: Eigenvalues and Eigenvectors in the Context of Finite Reflection Groups

In order to show that the vector $\underline{x}$ is an eigenvector of the matrix $B$ relating to the eigenvalue $\lambda,$ simply show that the product $B\underline{x}$ produces $\lambda\underline{x}$ as a result.

For your example this would mean that you should show that $Bx_{1} =x_{1}$ and that $Bx_{2} =-x_{2}$. (I assume that $x_{1}$ and $x_{2}$ should be column vectors ?)

You will need to make use of the trig identities for $\cos(A-B)$ and $\sin(A-B).$