1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quadratic forms, diagonalization

  1. Dec 20, 2008 #1
    Can a quadratic form always be diagonalised by a rotation??

    Thx in advance
  2. jcsd
  3. Dec 20, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Yes. That is because a quadratic form can always be written as a symmetric (hence self-adjoint) matrix. Thus, there always exist a basis consisting of orthogonal eigenvectors. Choosing your axes along those eigenvectors diagonalizes the matrix and, since the eigenvectors are orthogonal, that is a rotation.
  4. Dec 20, 2008 #3
    One thing to add to what Halls said: Not every orthogonal matrix is a rotation; there are reflections as well. That's not a big issue, since all you need to do is swap two of the axes to get the orientation right.

    Final point: Some quadratic forms cannot be written as a symmetric matrix over a field of characteristic 2; for example, x2 + xy + y2. (Since you're talking about rotations, you're probably working over the real numbers, where that's not an issue.)
  5. Dec 21, 2008 #4
    Great. Thx alot.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Quadratic forms, diagonalization
  1. Quadratic forms (Replies: 2)

  2. Quadratic forms (Replies: 1)

  3. Quadratic Forms (Replies: 5)