Eigenvectors of symmetric matrices

Can anyone prove that the eigenvectors of symmetric matrices are orthogonal?

HallsofIvy
Homework Helper
Let v be a eigenvector with eigenvalue $\lambda_1$ and u an eigenvector with eigenvalue $$\displaystyle \lambda_2$$, both with length 1.

$\lambda_1<v, u>= <\lambda_1v, u>$
(<u, v> is the innerproduct)
$= < Av, u>= \overline{<v, Au>}$
(because A is symmetric)
$= \overline{<v, \lambda_2u>}= \lambda<v, u>$
so that
$\lambda_1<v, u>= \lambda_2<v, u>$
$(\lambda_1- \lambda_2)<v, u>= 0$
Since $\lambda_1$ and $\lambda_2$ are not equal,
$\lambda_1- \lambda_2$ is not 0, <v, u> is.

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mathman