# expression of SVD of rank one matrix

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 $$He_1 = -u/\|u\|$$.