Quadratic forms, diagonalization

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Discussion Overview

The discussion centers on whether a quadratic form can always be diagonalized by a rotation, exploring the properties of quadratic forms, symmetric matrices, and the implications of orthogonal matrices.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant asserts that a quadratic form can always be diagonalized by a rotation because it can be represented as a symmetric matrix, which allows for a basis of orthogonal eigenvectors.
  • Another participant adds that while orthogonal matrices can diagonalize quadratic forms, not all orthogonal matrices represent rotations, as reflections are also included. They suggest that swapping axes can correct orientation.
  • A further point is raised regarding the limitation of quadratic forms that cannot be expressed as symmetric matrices over fields of characteristic 2, providing an example of such a form.

Areas of Agreement / Disagreement

Participants express differing views on the nature of orthogonal matrices and their relation to rotations, indicating that the discussion contains competing perspectives on the diagonalization of quadratic forms.

Contextual Notes

There is a mention of specific conditions under which certain quadratic forms cannot be represented as symmetric matrices, particularly in fields of characteristic 2, which may limit the general applicability of the claims made.

student111
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Can a quadratic form always be diagonalised by a rotation??

Thx in advance
 
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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.
 
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.)
 
Great. Thx a lot.
 

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