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If you have a symmetric matrix, is it orthogonally diagonalizable? Or is the converse only true?

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They are equivalent ,

$A$ being orthogonally diagnoizable you mean that there's an orthogonal matrix $U$ and a diagnonal matrix $D$ such that $A=UDU^{−1}=UDU^T$.

$A$ is then symmetric,( since $D$ is diagnonal, $D^T=D$)

$$A^T=(UDU^T)^T=(DU^T)^TU^T=UD^TU^T=UDU^T=A.$$

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    $\begingroup$ Even if there's no need to use TeX it's always better to use it for many reasons. $\endgroup$ – hjhjhj57 Nov 11 '14 at 7:00

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