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Homework Help: Diagonal bases in transformations

  1. Sep 7, 2011 #1
    1. The problem statement, all variables and given/known data

    Let T: R3 - R3 be the linear operator given by

    T = -y + z
    -x + z
    x + y

    Find a basis B' for R3 relative to which the matrix for T is diagonal using the standard basis B for R3.

    2. Relevant equations

    [T]B' = P-1[T]BP

    3. The attempt at a solution

    I find the standard matrix for T to be

    0 -1 1
    -1 0 1
    1 1 0

    The characteristic equation of which, I find to be

    (lambda)^3 -3(lambda) + 2 = 0

    Which has no real solutions? What can I do?


  2. jcsd
  3. Sep 7, 2011 #2


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    well the matrix is symmetric so that should ensure real eigenvalues...

    from visual inspection it appears 1 is a root
  4. Sep 7, 2011 #3


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    Science Advisor
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    Hi Derryck! :smile:

    (have a lambda: λ and try using the X2 icon just above the Reply box :wink:)
    erm :redface:

    how can a cubic equation have no real solutions? :wink:
  5. Sep 7, 2011 #4


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    I also got the same characteristic equation as well...
  6. Sep 7, 2011 #5
    Ok thanks guys. See the thing is I just put it into excel to help me find roots. I must have entered the wrong formula though:( It came up with an irrational number? Anyway...I can definitely see that 1 is a root now...thanks...
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