Suppose $A = uu^T, u \neq 0$. Then $A$ is a symmetric rank one matrix. Then, we can get $A = Q \Sigma Q$, where $Q^T=Q$ and $Q^TQ=I$. If $||u||_2 = 1$, we can rewrite the SVD as $A = Qe_1e_1^TQ$.
My question is what is the expression of $Q$.
One answer is $Q = \frac{2(u + ||u||_2 e_1)(u + ||u||_2 e_1)^T}{||u + ||u||_2 e_1||^2_2} - I$. How to get it?
$Q$ can be any orthogonal matrix whose first column is $u/\|u\|$. In other words, $Q$ needs to satisfy $Q e_1 = u/\|u\|$.
The matrix you have given is $Q = -H$, where $H$ is the unique Householder transformation satisfying $He_1 = -u/\|u\|$.