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Homework Help: Linear Algebra Question: Eigenvectors & Eignvalues

  1. Jul 17, 2008 #1
    This is my last week in Linear Algebra. I am working on our last homework assignment before the exam so I want to make sure I know what I am doing.

    In each part, make a conjecture about the eigenvectors and eigenvalues of the matrix A corresponding to the given transformation by considering the geometric properties of multiplication by A. Confirm each of your conjectures with computations.

    A. Reflection about the line y=x
    B. Contraction by a factor of 1/2

    I'm not quite sure how to get started.
     

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  3. Jul 18, 2008 #2

    HallsofIvy

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    reflection about a line does not change the length of a vector. That sharply restricts the possible eigenvalues. Also consider two crucial cases:
    (1) reflection about y= x of the vector <1, 1>, lying in y= x.
    (2) reflection about y= x of the vector <1, -1>, perpendicular to y= x.

    For the second, think about happens to <x, y>. "Contraction by a factor of 1/2" changes <x, y> to what vector? How can that be equal to [itex]\lambda <x, y>[/itex]?
     
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