Let x be a non-zero (column) vector in $R^n$. What is the necessary and sufficient condition for the matrix $A = I − 2xx^T$ to be orthogonal?
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Hint: $A$ is orthogonal if and only if $A.A^T=I$. Note that in your case $A^T=A$ and $x^Tx=\|x\|^2$. So, after some simplification, we have $AA^T=I+4(\|x\|^2-1)xx^T$. when the quantity on the right hand side be I?