Q:

Eigendecomposition of Householder matrix

I have already solved the first and second part of the question below. But, I am having a problem with the third part.
Let H be the Householder transformation
$H=I_m−2ww^T$, $‖w‖^2= 1$
and let $x ∈ R^n$ be a nonzero vector such that $x≠αe_1$. $e_1$ is the column of an identity matrix.
(a)  Show that $Hx=αe_1$, $|α|=‖x‖_2$
if $w= (x−αe_1)/β, β=‖x−αe1‖_2$.  
(b) What are the eigenvalues of H?
(c)  Show that the eigendecomposition of H may be written $H=QΛQ^T$ where Q is the Householder transformation such that $Qw=±e_1$. It does not matter which sign is chosen.

A:

Take
$$
\Lambda = I - 2e_1e_1^T = \pmatrix{-1\\&1\\&&\ddots\\&&&1}.
$$
Note that since $Q^{-1} = Q$ (i.e. $Q^2 = I$), we have $Q e_1 = \pm w$ (whichever sign you've chosen).  With this in mind, compute
$$
Q \Lambda Q^T = Q(I - 2e_1e_1^T)Q^T = QQ^T - 2(Q e_1)(Qe_1)^T
$$

