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Math Help - 2x2 matrix Q where Q^-1AQ is equal to a rotation followed by scaling a positive r.

  1. #1
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    2x2 matrix Q where Q^-1AQ is equal to a rotation followed by scaling a positive r.

    A = 4 2
    -5 -2

    Im suppose to find an invertible 2x2 matrix Q such that Q^-1AQ is equal to a rotation followed by scaling a positive real number, and write down both the rotation matrix and the scaling matrix. Also, what is the angle of the rotation matrix?

    When I do the work I get

    Q^-1AQ = -5 1
    -37 7

    I'm not sure if this is correct and if it is, I dont know how to find the scaling and rotation matrix
    Last edited by mr fantastic; January 31st 2011 at 05:57 PM. Reason: Title.
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  2. #2
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    Quote Originally Posted by Arita View Post
    A = 4 2
    -5 -2

    Im suppose to find an invertible 2x2 matrix Q such that Q^-1AQ is equal to a rotation followed by scaling a positive real number, and write down both the rotation matrix and the scaling matrix. Also, what is the angle of the rotation matrix?

    When I do the work I get

    Q^-1AQ = -5 1
    -37 7

    I'm not sure if this is correct and if it is, I dont know how to find the scaling and rotation matrix
    \displaystyle\text{Rotation Matrix}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{bmatrix}\begin{bmatr  ix}x\\y\end{bmatrix}

    \displaystyle\text{Scaling Matrix}=\begin{bmatrix}s_x&0\\ 0&s_y\end{bmatrix}\begin{bmatrix}x\\y\end{bmatri  x}
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  3. #3
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    I'm getting that

    Q^-1AQ = 1/-5 [0 1] [4 2] [-3 1]
    [-5 3] [-5 -2] [5 0]

    Which then works out to be

    [-1 1]
    [1 1]

    Then only problem im having is that if i take out 1 r then the matrix stays the same for the rotation and I dont think there is a angle associated with that rotation matrix.
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  4. #4
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    Quote Originally Posted by Arita View Post
    I'm getting that

    Q^-1AQ = 1/-5 [0 1] [4 2] [-3 1]
    [-5 3] [-5 -2] [5 0]

    Which then works out to be

    [-1 1]
    [1 1]

    Then only problem im having is that if i take out 1 r then the matrix stays the same for the rotation and I dont think there is a angle associated with that rotation matrix.
    Is Q a transition matrix?
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  5. #5
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    I'm not sure, the question just says to find the rotation matrix, the scaling matrix and what the angle of the rotation matrix is.
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  6. #6
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    Quote Originally Posted by Arita View Post
    I'm not sure, the question just says to find the rotation matrix, the scaling matrix and what the angle of the rotation matrix is.
    How did you obtain Q and then?
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