Quadratic forms, diagonalization

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 alot.
 

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