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for any symmetric real matrix $S$, the following eigendecomposition exists:

$$ S = Q \Lambda Q^{\top} $$

where $Q$ is a unitary matrix, consisting of the eigenvectors of $S$ wikipedia . By definition of unitary, we have $Q^{\top}Q=QQ^{\top}=I$. Given an orthonormal set of eigenvectors, $Q^{\top}Q=I$, is trivial. How can one show $QQ^{\top}=I$ ?

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If $Q^TQ = I$, then $Q^T = Q^{-1}$. Now $QQ^{-1} = I$, so... –  Rahul Sep 22 '11 at 0:15
    
it is a full rank matrix and its inverse is $Q$. I don't get your question. –  user13838 Sep 22 '11 at 0:15
    
You might find this useful; it explains why $AB=I$ implies $BA=I$ for general square matrices $A$ and $B$. –  Nick Strehlke Sep 22 '11 at 0:21
    
Thanks All. it simply answered my question. I was not thinking about it that way. –  Austin Sep 22 '11 at 1:06
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2 Answers

up vote 1 down vote accepted

If $\mathbf Q^\top\mathbf Q=\mathbf I$, then $\mathbf Q\mathbf Q^\top\mathbf Q=\mathbf Q$. Then consider postmultiplying both sides by $\mathbf Q^{-1}$...

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Hmm, I think, the more interesting part in the original question is in the direction: "why is it with a symmetric matrix S, that the diagonalization $ \small S = Q \Lambda Q^{-1} $ provides a unitary matrix Q , such that $ \small Q^T = Q^{-1} $ ?" Which, of course, can be answered by considering the equality of S with its transpose: $\small S = S^T=(Q^T)^{-1} \Lambda Q^T = Q \Lambda Q^{-1} \to Q=(Q^T)^{-1} $ and $ \small Q^{-1}=Q^T $

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