Deriving complex vector and scalar to satisfy this Householder transform equation

Question

I'm currently really struggling with part b. of this problem:

Letting $H(v) = I - \frac{2v v^*}{v^* v}$ for some complex vector, v, and where $v^*$ is the conjugate transpose of v.

So for part b, I know that we can constrain $\delta$ by realizing that $|\delta| = 1$, since H is unitary, and thus, will have a complex 2-norm of 1.

I also realize that $H(v)x = \delta y$ can be expressed as $\gamma v = x - \delta y$, where $\gamma$ is a complex constant, and that $H(\gamma v) = H(v)$.

So now I believe I'm at the point where I need to guess/determine what value of $\delta$ I should use so that:

1. $|\delta|$ = 1
2. $\gamma v = x - \delta y$
3. $H(\gamma v) = H(v) = H, s.t. Hx = \delta y$, where $||x||_2 = ||y||_2$.

I've tried several complex numbers for $\delta$ such that $|\delta| = 1$, but nothign has worked out for me when I actually use that value to determine v, then H, and then plug everything in ;(

I have a feeling I need to make use of inner-product spaces concepts, but I'm at such a loss at the moment.

Any help with how I should derive the formulas for $v_c$ and $\delta$ would be greatly appreciated!

Answer 1

Turns out that, if we let $\delta = \frac{(y^*) x}{|(x^*) y|}$, and $v = x - \delta y$, then $H(v)x = \delta y$ if $||x||_2 = ||y||_2$