This is intended to be a more generic question not about a specific system. Given a hermitian matrix $H(x_1,\dots,x_n)$ depending non-linearly on some real parameters $x_1,\dots,x_n$. We want these to be determined numerically such that $$H=0$$ and assume we already know that a set of parameters always exists such that this holds.
Would it be more efficient to directly solve the above equation or to optimise its Frobenius norm $||H||_F^2$ (which would just be a scalar)? Also, which method would you apply?