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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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?

We have $AA^T = (I − 2xx^T)(I − 2 x x^T)^T = (I − 2xx^T)(I − 2 x x^T) = I -4xx^T + 4xx^T xx^T$. Let $X = xx^T$. Then $AA^T = I$ iff $X = X^2$, or $X$ has the minpoly $\lambda^2 - \lambda$. (edit: better use Tapu's answer). That is, the condition is: $X = \| x \|^2 X$ where $X \neq 0$.

Tapu · Accepted Answer · 2012-09-01 18:07:30Z

up vote 1 down vote accepted

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?

answered Sep 1 '12 at 18:07

Tapu
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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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