Letube an unit vector in $\displaystyle R^{n}$ and $\displaystyle H=I-2*u*u^{T}$. Show that H is orthogonal, symmetric, and its own inverse.

Assume H is orthogonal then $\displaystyle H^{-1}=H^{T}$ and $\displaystyle HH^{T}=I$

$\displaystyle [(I-2*u*u^{T})(I-2*u*u^{T})^{T}]^{T}$

$\displaystyle (I-2*u*u^{T})(I-2*u*u^{T})^{T}$

Now we have $\displaystyle H*H^{T}=I$

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