Quadratic forms, diagonalization

[student111]

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

Thx in advance

 

[HallsofIvy]

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.

 

[adriank]

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.)

 

[student111]

Great. Thx a lot.