Quadratic forms, diagonalization

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A quadratic form can always be diagonalized by a rotation because it can be represented as a symmetric matrix, which guarantees the existence of orthogonal eigenvectors. By aligning the axes with these eigenvectors, the matrix can be diagonalized through rotation. While not all orthogonal matrices represent rotations—some are reflections—adjusting the axes can correct the orientation. However, it's important to note that certain quadratic forms cannot be expressed as symmetric matrices in fields of characteristic 2. Overall, for real numbers, diagonalization via rotation is always possible.
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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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