Minimizing a quadratic equation, with constraint.

Question

I have a problem, I want to minimize this: $$ \min_{w} {w^\dagger H_1 w}\\ s.t. {w^\dagger H_2 w} = \mu^2 \\ ||w||^2 = 1 $$ with $\mu$ being real positive number, and $H_1, H_2$ are matrices with same dimensions.

I know the solution if $\mu=0$, but I need this for a positive $\mu$. The result will be something depending on $\mu$ right?

Thank you in advance.

Answer 1

Note that $<H_2w,w>=\mu^2=\mu^2<w,w>$ so that $<(H_2-\mu^2 I)w,w>=0$, so you can assume that your conditions are $<(H_2-\mu^2 I)w,w>=0$ and $<w,w>=1$ and then you can use your solution for $\mu=0$ .