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## Eigenvalues and Eigenvectors of Symmetric Real Matrices |

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z | overview |

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

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automatically generated 4/28/2005 |