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Math Help - Orthonormal Set of Eigenvectors of A

  1. #1
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    Orthonormal Set of Eigenvectors of A

    Problem:



    Part (a): This part seems simple. I just find the eigenvectors of the matrix A, use the GS process, and then normalize to get an orthonormal set. Is this correct?

    Part (b): I'm not sure how to do this part. Is "S" referring to [S]B? That is, do I use the eigenvectors of A=[S]B? I'm pretty lost here...

    Part (c): Not sure how to do this without completing part b.



    Any help will be appreciate and I'll click the thank you button for you! Thanks!
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    MHF Contributor Drexel28's Avatar
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    Re: Orthonormal Set of Eigenvectors of A

    Quote Originally Posted by divinelogos View Post
    Problem:



    Part (a): This part seems simple. I just find the eigenvectors of the matrix A, use the GS process, and then normalize to get an orthonormal set. Is this correct?

    Part (b): I'm not sure how to do this part. Is "S" referring to [S]B? That is, do I use the eigenvectors of A=[S]B? I'm pretty lost here...

    Part (c): Not sure how to do this without completing part b.



    Any help will be appreciate and I'll click the thank you button for you! Thanks!
    I'm sorry, is this for a takehome exam?
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    Re: Orthonormal Set of Eigenvectors of A

    Quote Originally Posted by Drexel28 View Post
    I'm sorry, is this for a takehome exam?
    Nope. It's a study guide for the final. I can send you the whole thing if you need to verify that. Thanks!
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Orthonormal Set of Eigenvectors of A

    (a) The eigenvalues of A are \lambda=1 (double) and \lambda=1 (simple). We have \ker (A-I) \equiv \begin{Bmatrix}-x_1+x_3=0\\x_1-x_3=0\end{matrix} and an orthogonal basis is \{(1,0,1)^t,(0,1,0)^t\} .

    Now, \ker (A+I) \equiv \begin{Bmatrix}x_1+x_3=0\\2x_2=0\\x_1+x_3=0\end{ma  trix} and a basis is \{(1,0,-1)^t\} .

    Hence, B=\{(1/\sqrt{2},0,1/\sqrt{2})^t,(0,1,0)^t,(1/\sqrt{2},0,-1/\sqrt{2})^t\} is an orthonormal basis of eigenvectors of A .

    (b) (1/\sqrt{2},0,1/\sqrt{2})^t\equiv (1/\sqrt{2})\vec{u_1}+(1/\sqrt{2})\vec{u_3}=\ldots

    Hope you can continue.
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