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Math Help - Show that T is diagonalizable

  1. #1
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    Show that T is diagonalizable

    Suppose T: V----> V has only two distinct eigenvalues k1 and k2

    Show that T is diagonalizable iff V = Ek1+ Ek2
    Last edited by dave52; March 12th 2013 at 11:43 AM.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Show that T is diagonalizable

    Quote Originally Posted by dave52 View Post
    Suppose T: V----> V has only two distinct eigenvalues 1 and 2

    Show that T is diagonalizable iff V = E1 + E2
    Hi dave52!

    It seems to me that it is not true.

    \begin{bmatrix}1&0&0 \\ 0&1&0 \\ 0&0&2\end{bmatrix}

    This matrix is diagonal, has distinct eigenvalues 1 and 2, but \mathbb R^3 \ne E_1 + E_2.
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  3. #3
    Senior Member jakncoke's Avatar
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    Re: Show that T is diagonalizable

    I think you meant to say that your vector space has Dimension 2.
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    Re: Show that T is diagonalizable

    sorry I just fixed the problem
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  5. #5
    Super Member ILikeSerena's Avatar
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    Re: Show that T is diagonalizable

    Let's first proof T is diagonalizable \Rightarrow V = E1 + E2.

    If T is diagonalizable, it can be written as PDP^{-1}, where D is a diagonal matrix with its eigenvalues on the main diagonal, and P represents a basis of eigenvectors.
    That basis spans V.
    Therefore V is the direct sum of the eigen spaces.


    Now let's proof V = E_1 + E_2 \Rightarrow T is diagonalizable.

    This means that that basis of E_1 together with the basis of E_2 span V.
    Put those bases next to each other in a matrix P, and put the eigenvalues diagonally in a corresponding matrix D, and you have that

    TP = PD

    T = PDP^{-1}

    So T is diagonalizable.


    Therefore T is diagonalizable iff V = E_1+ E_2 \qquad \blacksquare
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