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Math Help - Eigenvalues and Eigenvectors in the Context of Finite Reflection Groups

  1. #1
    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

     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
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    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).
    Thanks from Bernhard
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