1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.