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How to classify a quadratic surface?

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

    Classify the quadratic surface:

    2x^2 + 4y^2 - 5z^2 + 3xy - 2xz + 4yz = 2

    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

    A = [2, 3/2, -1]
    [3/2, 4, 2]
    [-1, 2, -5]

    Then, find the eigenvectors of A-tI...check how many are positive, negative, or 0 and classify using the information? Is that the right way to proceed?
     
  2. jcsd
  3. Mar 22, 2009 #2
    I think you have the right idea, but the wrong terminology: You are trying to find the eigenvalues of A by finding zeros of the polynomial det(A-tI).
     
  4. Mar 22, 2009 #3
    Oh ok. And then, the fact that F(x, y, z) is = 2 doesn't make a difference, right? Because, it is the j that F(x, y, z) can be set equal to such that (x^T)*A*x = j?
     
  5. Mar 22, 2009 #4
    The sign of the RHS does make a difference, in this case it tells you how many "sheets" the quadratic surface has (have you found the eigenvalues yet?).
     
  6. Mar 22, 2009 #5
    I get two positive and one negative eigenvalues: -5.647, 1.661, 4.986. This would be a hyperboloid of one sheet.
     
  7. Mar 22, 2009 #6
    That's correct. :smile:
     
  8. Mar 22, 2009 #7
    Thanks a bunch!
     
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