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

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    Super Member Bernhard's Avatar
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    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

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

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

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

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

    I would appreciate help with this problem.

    Peter
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    Re: Eigenvalues and Eigenvectors in the Context of Finite Reflection Groups

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

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

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