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Math Help - Eigenvalues and Eigenvectors

  1. #1
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    Eigenvalues and Eigenvectors

    Let L be the line through the origin of R2 that makes an angle of ∏/4 with the positive x-axis, and let A be the standard matrix for the reflection of R2 about that line. Make a conjecture
    about the eigenvalues and eigenvectors of A and confirm your conjecture by computing them in the usual way.
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  2. #2
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    Re: Eigenvalues and Eigenvectors

    Seeing that I know what is going to happen, it's best for you to make the conjecture yourself. Draw a picture and see what happens to arbitrary vectors. Are any vectors invariant under the reflection (i.e. do they stay the same)? Are any vectors scaled, but otherwise lie on the same line spanned by itself?
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  3. #3
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    Re: Eigenvalues and Eigenvectors

    I think the eigenvalues of A are λ=1 and λ=-1 with corresponding eigenspaces?
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  4. #4
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    Re: Eigenvalues and Eigenvectors

    Yes, sounds about right. Nevertheless, you should check it as suggested by your question. Do you know what the matrix for A is? Hint: the matrix depends entirely on where you take the basis vectors (0,1) and (1,0).
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