Give the conditions on the $m\times1$ vector so that a matrix $H$ is orthogonal

Question

Give the conditions on the $m\times 1$ vector $x$ such that the matrix $H=I_m-2xx'$ is orthogonal.

The only solution I have found and verified is $x$ could be the zero vector; however, I know there should be more than just this one solution.

Thank you for your time.

Answer 1

The definition of orthogonal matrix $H$ is $HH^T=I$

$$ I=(I-2xx^T)(I-2xx^T)^T=(I-2xx^T)(I-2xx^T) = I-4xx^T+4xx^Txx^T = \\I-4xx^T+4x(x^Tx)x^T = I+4(x^Tx-1)xx^T. $$

So either $xx^T=0$ (solution that you have found), or $x^Tx=|x|^2=1$, which means length of vector $x$ is unity.

Answer 2

I presume $x'$ is the transpose of $x$. Then $H$ is symmetric, so it's also orthogonal iff $H^2=I$. But $$H^2=(I-2xx')^2=I-4xx'+4xx'xx'=I-4(1-a)xx'$$ where $a=x'x$ is a scalar. So the matrix $H$ is orthogonal iff $a=1$.