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Math Help - Find the values of "z" for which this Matrix has no real eigenvalues

  1. #1
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    Find the values of "z" for which this Matrix has no real eigenvalues

    I got this problem:

    Find the values of "z" for which this Matrix has no real eigenvalues.
    The Matrix is the following:
    [ 1 -1]
    [ 1 z]

    I tried to solve this problem as follows:
    1) I calculated the eigenvalues for the Matrix keeping "z" as a constant
    2) I got:
    -1/z1/2 and 1/z1/2
    So if I substitute z with >0 I have real eigenvalues.
    For z <0 I don't have real eigenvalues.

    But what happens with the 0? If I substitute 0 for z the result is not defined, but if I substitute 0 for z in the matrix I get two real eigenvalues (0,0).

    Therefore I say that the Matrix has no real eigenvalues for z<0

    Is this true?
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  2. #2
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    Re: Find the values of "z" for which this Matrix has no real eigenvalues

    Some thing is going wrong!

    The eigenvalues are the sol of the following equation
    (1-\lambda)(z-\lambda)+1=0
    that is
    \lambda^2-\lambda(1+z)+(1+z)=0
    so this equation has a real valued solutions if and only if
    (1+z)^2-4(1+z)\geq0
    which equivalently means
    -1\leq z\leq3.
    Thanks from infernalmich
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  3. #3
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    Re: Find the values of "z" for which this Matrix has no real eigenvalues

    But if I set for example z=5, than I get real eigenvalues: (1,2679 4,7321) - therefore it has real eigenvalues also for z=5 or not?
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  4. #4
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    Re: Find the values of "z" for which this Matrix has no real eigenvalues

    Sorry! the eigenvalues are real when z is out of the interval -1<z<3.
    So, we have real eigenvalues if and only if z\geq3 or z\leq-1
    Thanks from infernalmich
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  5. #5
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    Re: Find the values of "z" for which this Matrix has no real eigenvalues

    Kmath corrected while I was typing this!
    Last edited by HallsofIvy; June 14th 2012 at 10:08 AM.
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