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Matrices, Proof and Eigenvalues.

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Looking for some help with the proof if possible.

    Vector r =
    x
    y
    z

    Rotation R =
    cos(θ) 0 sin(θ)
    0 1 0
    -sin(θ) 0 cos(θ)

    r' = Rr

    It asks me to prove that
    r'.r' = r.r




    Second part of the question is about eigenvalues, it asks me to find the three eigenvalues of R.
    I used the formula
    det(m - λI)

    Where m = R, λ = the eigenvalues and I is the appropriate identity matrix

    I end up with λ = cos() or 1, which is clearly wrong as there isn't enough answers.
    Somebody in class mentioned imaginary numbers, but I'm unsure as to how to proceed.


    3. The attempt at a solution

    First Part:

    I found r' to be
    xcos(θ) + xsinθ
    y
    -zsin(θ) + xcos(θ)

    To get r'.r' am I right to just multiply two of the above together, as in
    (xcos(θ) + xsin(θ))(xcos(θ) + xsin(θ))
    (yy)
    (-zsin(θ) + xcos(θ))(-zsin(θ) + xcos(θ))

    Because this is the way I did it and it doesn't lead to the same answer.

    Obviously these should all have big brackets around them, but I am unsure of how to represent them on here, if someone would advise me I would gladly fix that.

    Second Part:
    I used the formula
    det(m - λI)

    Where m = R, λ = the eigenvalues and I is the appropriate identity matrix

    I end up with λ = cos() or 1, which is clearly wrong as there isn't enough answers.
    Somebody in class mentioned imaginary numbers, but I'm unsure as to how to proceed.
     
  2. jcsd
  3. Mar 18, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    This is incorrect but I expect it is a typo.

    Yes, that is what you want to do- and add them. You need to fix that first coordinate of course. What did you get? Don't forget that [itex]cos^2(\theta)+ sin^2(\theta)= 1[/itex].

    We're not mindreaders! No one can tell you what you did wrong unless you tell us what you did!
     
  4. Mar 18, 2012 #3
    What's wrong with the first bit, have I made a stupid mistake?

    I've attached what I have done.
     

    Attached Files:

  5. Mar 18, 2012 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You have [itex]det(A-\lambda I)= (cos(\theta)- \lambda)(1- \lambda)(cos(\theta)- \lambda)[/itex] which is wrong. That is the product of the values on the main diagonal but is not the determinant.
     
  6. Mar 19, 2012 #5
    Ah yeah, I've sorted that all out now.
    Thank you so much for the help mate.
     
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