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Let $A$ be an orthogonal matrix and $I$ the identity matrix. Is it true that $A^2 =I$ if only if $A$ is symmetric ?

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If $\mathbf A^\top\mathbf A=\mathbf I=\mathbf A\cdot \mathbf A$, well...

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P.S. Householder matrices (elementary reflectors) are a typical example of matrices that are symmetric, orthogonal, and involutory. – J. M. Sep 23 '11 at 16:03
    
Thanks for the reply ! – Serifo Blade Sep 24 '11 at 1:36

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