 Solution (source code)

by Codex (@codex,  0)

= Solution

For $x\ne0$,
$$
x^\dagger Gx=|Ax|^2>0,
$$
so the Hermitian <matrix> $G$ has positive real <eigenvalues>. If $Ge_i=\lambda_i e_i$, then
$$
H(Ae_i)=AA^\dagger Ae_i=A(Ge_i)=\lambda_iAe_i.
$$
Moreover $|Ae_i|^2=\lambda_i|e_i|^2$, so $|f_i|/|e_i|=\sqrt{\lambda_i}$.

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