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Another eigenvalue problem.

  1. Mar 31, 2008 #1
    Find a 2\times 2 matrix A for which

    E_4 = span [1,-1] and E_2 = span [-5, 6]

    where E_(lambda) is the eigenspace associated with the eigenvalue (lambda)


    relevant equations: Av=(lambda)v

    3. The attempt at a solution

    I've pretty much gotten most of the eigenspace/value problems down, but this one I'm clueless on how to work. Seems that you would have to work backwards , but I don't know how to do that.
     
  2. jcsd
  3. Mar 31, 2008 #2

    CompuChip

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    Homework Helper

    Suppose you have a matrix A and it is diagonalizable. That means you can write it as
    [tex]A = C D C^{-1}[/tex]
    When A is given, how can you construct the matrices C and D?

    Once you answer this question, you'll also see the answer to your problem.
     
  4. Mar 31, 2008 #3

    HallsofIvy

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    You can just do it directly. You are told that
    [tex]\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}1 \\ -1\end{array}\right)= 4\left(\begin{array}{c}1\\ -1\end{array}\right)= \left(\begin{array}{c}4\\ -4\end{array}\right)[/tex]
    which gives you two equations for a, b, c, d and
    [tex]\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}-5 \\ 6\end{array}\right)= 2\left(\begin{array}{c}-5\\ 6\end{array}\right)= \left(\begin{array}{c}-10\\ 12\end{array}\right)[/tex]
    which gives you another two equations. Solve those 4 equations.
     
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