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Explain how to modify the definition of the Householder reflector so that it works for a vector with complex entries. That is, given $x \in \mathbb C^m$ , how should be define the unit vector $v$ so that $(I_d−2vv^*)x = \|x\|e_1$?

I know $\mathbb C^m$ is an actually 2m dimensional space, so how to modify the above formula in order to suit for $\mathbb C^m$ case?

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The modification goes as follows \begin{align} \hat{u} &= \frac{x + e^{i\theta} ||x||e_1}{||x + e^{i\theta} ||x||e_1||}, \\ H &= -e^{-i\theta}(I - 2 \hat{u} \hat{u}^*), \end{align} where $x_1=e^{i\theta}|x_1|$.

In this case, you have $Hx = ||x|| e_1$.

See https://arxiv.org/pdf/math-ph/0609050.pdf at page 19 for more details.

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This is discussed in lots of textbooks on numerical linear algebra. See for example:

http://www.cs.utexas.edu/users/rvdg/books/HQR.pdf

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  • $\begingroup$ The material that you provided works for real vector but I want an answer for complex vector. $\endgroup$
    – Gatsby
    Sep 18, 2016 at 15:57
  • $\begingroup$ This will work in the complex case as well. Keep in mind that for complex $z$, $\mbox{sign}(z)=z/|z|$. $\endgroup$ Sep 18, 2016 at 16:26
  • $\begingroup$ I'm thinking about a whole day but don't understand what you said. Can you add more details? $\endgroup$
    – Gatsby
    Sep 19, 2016 at 18:55
  • $\begingroup$ I'm afraid that you're going to have to work through the calculations- try it on a random example and see what happens. $\endgroup$ Sep 19, 2016 at 20:04

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