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Diagonal Quadratic Forms of a Matrix

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Let the quadratic form F(x,y,z) be given as

    F(x,y,z) = 2x^2 + 3y^2 + 5z^2 - xy -xz - yz.

    Find the transitional matrix that would transform this form to a diagonal form.


    2. Relevant equations

    A quadratic form is a second degree polynomial equation in three variables which has the form

    F(x,y,z) = ax^2 + by^2 + cz^2 + 2dxy + 2exz + 2fyz

    where coefficients a through i are real numbers.

    The curve F(x,y,z) = j can be written in the form (x^T)*A*x = j where

    x = [x, y, z]

    A = [a, d, e]
    [d, b, f]
    [e, f, c]

    3. The attempt at a solution

    I need to find the eigenvectors of (A-sI) to form the transitional matrix. Does (x^T)*A*x = j have anything to do with finding this matrix? Having trouble picturing all of this...
     
  2. jcsd
  3. Mar 22, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Don't worry about "(x^T)*A*x= j", that will come automatically once you have found the correct transition matrix. First, find the eigenvalues for your matrix. Then find the eigenvectors corresponding to each eigenvalue. Since it is symmetric,there will exist a "complete set" of eigenvectors- a basis for the space consisting of eigenvectors. P will be the matrix having those eigenvectors as columns, P-1 its inverse. Then P-1AP= D, a diagonal matrix having the eigenvalues on the main diagonal.
     
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