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Homework Help: Finding Linear Transformation that will remove cross product term.

  1. Jun 29, 2011 #1
    1. The problem statement, all variables and given/known data

    Find a linear transofmration from X={x1,x2,x3} to U={u1,u2,u3} which will remove the cross product term in the quadratic form of equation 2X12+4X22+5X32-4X1X3
    and thus write the resulting quadratic form in u1,u2,u3.
    2. Relevant equations



    3. The attempt at a solution
    No idea at the moment. I presume that the 4x1x3 term is the term that I need to have disappear, but other than that, I can sure use some help. Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 30, 2011 #2

    vela

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    Let
    [tex]\vec{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}[/tex]
    Find the matrix A such that
    [tex]\vec{x}^\mathrm{T} A \vec{x} = 2x_1^2+4x_2^2+5x_3^2-4x_1x_3[/tex]
    You want to diagonalize this matrix.
     
  4. Jun 30, 2011 #3
    Hello vela.
    Thanks for the reply!
    The problem actually gave a matrix for a previous part so just diagonalizing it solved the problem fairly easily. However, if there are no given matrix, how would I find the matrix?
     
  5. Jun 30, 2011 #4

    vela

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    Try a 2x2 example. Calculate
    [tex]\begin{pmatrix} x & y \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}[/tex]
    Compare it to [itex]Ax^2 + Bxy + Cy^2[/itex]. What values would you choose for a, b, c, and d to get the coefficients A, B, and C? Note that you're shooting for a symmetric matrix. Can you generalize this to the 3x3 case?
     
  6. Jun 30, 2011 #5
    I see. I understood it now! Thanks a lot for your help :)
     
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