How to solve the system $$x^\dagger A_ix=0$$ $$x^\dagger x=1$$ where $x\in\mathbb{C}^n$ and $\{A_1, A_2,... A_m\}, A_i\in\mathbb{C}^{n\times n}$ is a given set of $m>n$ complex symmetric matrices?
We can write $x^\dagger A_ix=0 \implies x\in Ker(A_i)$ or $x \perp A_ix$ but I have hard time figuring out what the second case means. I also looked into quadratic programming libraries but I don't see how to deal with the fact that we have a system of equations and not only one quadratic form.