Quadratic forms, diagonalization

In summary, a quadratic form can always be diagonalised by a rotation because it can always be written as a symmetric matrix and a basis of orthogonal eigenvectors can be chosen. However, not all orthogonal matrices are rotations and some quadratic forms cannot be written as symmetric matrices over fields of characteristic 2.
  • #1
student111
16
0
Can a quadratic form always be diagonalised by a rotation??

Thx in advance
 
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  • #2
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.
 
  • #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.)
 
  • #4
Great. Thx alot.
 

1. What is a quadratic form?

A quadratic form is a mathematical expression that contains terms raised to the second power, such as x^2, y^2, or xy. It can be written in the form of ax^2 + by^2 + cxy, where a, b, and c are constants.

2. How is diagonalization related to quadratic forms?

Diagonalization is a process of transforming a quadratic form into a simplified form that only contains diagonal terms, with all other terms equal to 0. This simplifies the form and makes it easier to solve for its properties.

3. What is a diagonal matrix?

A diagonal matrix is a special type of matrix where all the elements outside of the main diagonal (the diagonal from the top left to bottom right) are equal to 0. This is the result of diagonalization, where a quadratic form is transformed into a simplified form with only diagonal terms.

4. How is diagonalization useful in mathematics?

Diagonalization is useful in mathematics because it simplifies a quadratic form and makes it easier to solve for its properties, such as its eigenvalues and eigenvectors. It also allows for easier manipulation and calculation of the form, making it a useful tool in various areas of mathematics, including linear algebra, optimization, and statistics.

5. Can any quadratic form be diagonalized?

Yes, any quadratic form can be diagonalized through a series of transformations, such as completing the square and using linear transformations. However, the process may not always be straightforward and may require advanced mathematical techniques to fully diagonalize the form.

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