Mathematics-Online lexicon: Eigenvalues and Eigenvectors of Symmetric Real Matrices

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	Mathematics-Online lexicon:
	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 be a real symmetric $n \times n$ - matrix.

1.

All eigenvalues of

are real.

2.

Eigenvectors to different eigenvalues are orthogonal.

3.

$\mathbb{R}^n$ has an orthonormal basis consisting of eigenvectors of

4.

Let $\{v_1, \ldots , v_n \}$ be an orthonormal basis consisting of eigenvectors of

. The matrix

, whose columns are $v_1, \ldots , v_n$ , i.e

has the form

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

is orthogonal, i.e..

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

5.

Let $\{v_1, \ldots , v_n \}$ be an orthonormal basis out of eigenvectors of

, let $T = (v_1, \ldots , v_n)$ and denote by $\lambda_i$ the eigenvalue corresponding to

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 as column vectors of

automatically generated 4/28/2005