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Thread: Normal orperator and eigenvalue

  1. #1
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    Normal orperator and eigenvalue

    Let $\displaystyle V=R^2 $ and define $\displaystyle T\in L(V) $ as the 90-degree rotation operator given by $\displaystyle T(x,y)=(-y,x) $ where $\displaystyle x,y \in R $ are the components of a vector in V.
    a- show that T is normal
    b-T does not have any eigenvalues in R.
    Thanks
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    For a), find the matrix which represents $\displaystyle T$ with respect to the standard basis. Then show that the matrix is normal.

    For b), suppose that $\displaystyle T(x,y)=\lambda(x,y)=(-y,x)$. What can you tell about $\displaystyle \lambda$? (Equivalently, find the roots of the characteristic polynomial of the matrix you found in part a) ).
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