For a given vector $u$ in $\mathbb R^n$ with unit norm $\|u\|_2=1$ let $$H=I-2uu^T$$

$a)$ Determine all eigenvalues and associated eigenvectors of $H$

My attempt was to assume $u=\begin{bmatrix}{\frac{1}{\sqrt2}}\\{\frac{1}{\sqrt2}}\end{bmatrix}$ and then calculate $uu^T$, then evaluate $H$ and then its eigenvalues so on but obviously my assumption is incorrect so how should I go about it?

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$b)$ For a given $x \in \mathbb R^n$ how are $x$ & $Hx$ related to each other?

This one I simply don't understand…

Any help with the first one since perhaps it is the key to understanding the second one.

## Best Answer

Calculate $H^2$. It’s identity. Hence eigenvalues are $\pm1$.