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# Eigenvalues and Eigenvectors of Symmetric Real Matrices

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Symmetric matrices over are normal. Thus they are diagonizable by a unitary transformation. With respect to the principal axis transformation of quadrics this property is of fundamental interest. Therefore we give a seperate summary of the properties of eigenvalues and eigenvectors of real symmetric matrices.

Let be a real symmetric - matrix.

1.
All eigenvalues of are real.
2.
Eigenvectors to different eigenvalues are orthogonal.
3.
has an orthonormal basis consisting of eigenvectors of
4.
Let be an orthonormal basis consisting of eigenvectors of . The matrix , whose columns are , i.e has the form

is orthogonal, i.e..

5.
Let be an orthonormal basis out of eigenvectors of , let and denote by the eigenvalue corresponding to Then is a diagonal matrix of the form

Note that the eigenvalues in the main diagonal are in the same ordering as the corresponding eigenvectors as column vectors of