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Math Help - Orthogonal Change of Variables

  1. #1
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    Orthogonal Change of Variables

    Find an orthogonal change of variables X=PY such that:

    2x1^2 + 2x1x2 - 2x2^2

    takes the form:

    Ay1^2 + By2^2

    Sorry if this may be confusing. The numbers next to the variables should be subscripts. For instance x1^2 should be x subscript 1 to the power of 2.
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  2. #2
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    Write the quadratic form as -2x_1^2+2x_1x_2-2x_2^2 = \begin{bmatrix}x_1&x_2\end{bmatrix} \begin{bmatrix}-2&1\\1&-2\end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix}. Then find the eigenvalues and eigenvectors of the matrix and diagonalise it.
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  3. #3
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    confused about the same type of problem

    my problem is find an orthogonal change in variables x=py such that
    4x1^2 + 10x1x2 + 4x2^2
    takes the form ?x1^2 + ?x2^2
    p=?
    the question marks are blanks for answers, there should be subscripts and ^2 is squared

    so i think i understand how to find p... you diagonalize and find the eigenvectors, but i dont know how to answer the x1^2 + x2^2 part.
    Last edited by luccasaurus; December 1st 2008 at 06:39 PM. Reason: forgot a part
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  4. #4
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    Quote Originally Posted by luccasaurus View Post
    my problem is find an orthogonal change in variables x=py such that
    4x1^2 + 10x1x2 + 4x2^2
    takes the form ?x1^2 + ?x2^2
    p=?
    the question marks are blanks for answers, there should be subscripts and ^2 is squared

    so i think i understand how to find p... you diagonalize and find the eigenvectors, but i dont know how to answer the x1^2 + x2^2 part.
    Step 1: Write the quadratic form as \mathbf{x}^{\textsc{t}}A\mathbf{x}, where \mathbf{x} = \begin{bmatrix}x_1\\x_2\end{bmatrix} and A = \begin{bmatrix}4&5\\5&4\end{bmatrix}.

    Step 2: Diagonalise A, to get  A = P^{\textsc{t}}DP, where D is the diagonal matrix \begin{bmatrix}-1&0\\0&9\end{bmatrix} (-1 and 9 being the eigenvalues of A), and P is the orthogonal matrix whose columns are the corresponding normalised eigenvectors.

    Step 3: Then the quadratic form is \mathbf{x}^{\textsc{t}}P^{\textsc{t}}DP\mathbf{x} = \mathbf{y}^{\textsc{t}}D\mathbf{y} = -y_1^2 + 9y_2^2, where \mathbf{y} = \begin{bmatrix}y_1\\y_2\end{bmatrix} = P\mathbf{x}.
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