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Eigenvalues and Eigenvectors of Symmetric Real Matrices

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Symmetric matrices over $ \mathbb{R}$ 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 $ A$ be a real symmetric $ n \times n$ - matrix.

All eigenvalues of $ A$ are real.
Eigenvectors to different eigenvalues are orthogonal.
$ \mathbb{R}^n$ has an orthonormal basis consisting of eigenvectors of $ A .$
Let $ \{v_1, \ldots , v_n \}$ be an orthonormal basis consisting of eigenvectors of $ A$. The matrix $ T$, whose columns are $ v_1, \ldots , v_n$, i.e $ T$ has the form

$\displaystyle T = (v_1, \ldots , v_n) ,$

is orthogonal, i.e..

$\displaystyle T^{-1} = T^{\operatorname t}.$

Let $ \{v_1, \ldots , v_n \}$ be an orthonormal basis out of eigenvectors of $ A$, let $ T = (v_1, \ldots , v_n)$ and denote by $ \lambda_i $ the eigenvalue corresponding to $ v_i .$ Then $ T^{\operatorname t}A T $ is a diagonal matrix of the form

$\displaystyle T^{\operatorname t}A T = D (\lambda_1, \ldots, \lambda_n) .

Note that the eigenvalues $ \lambda_i $ in the main diagonal are in the same ordering as the corresponding eigenvectors $ v_i$ as column vectors of $ T .$


  automatically generated 4/28/2005