Given $A\in\mathbb{C}^{n\times n}$, denote set $$S_1:=\{(\Re (x^*Ax),\Im(x^*Ax)) \mid\|x\|_2=1\}.$$ This set is called numerical range and it contains all eigenvalues of $A$. I found a program that can plot this set for any $A$. In this program I just enter matrix $A$.
Now, using that program I want to plot another set $$S:=\{(\Re (x^*Ax),\Im(x^*Ax))\mid \|x\|_2=1,\|Ax\|_2=1\}.$$ As you can see we have additional constraint $\|Ax\|_2=1$.
Denote set $$S_2:=\{(\Re (x^*Ax),\Im(x^*Ax))\mid\|Ax\|_2=1\}.$$
Conjecture: $S=S_1\cap S_2$.
If that conjecture holds, my idea was that I can use my current program to plot $S_1$ and $S_2$ and look to their intersection to find $S$. But still I need to do change of variables to be able to plot $S_2$:
Lets do SVD: $A=U\Sigma V^*$, where $\Sigma=\mathrm{diag}\{\sigma_1,\ldots,\sigma_n\}$, then $$S_2:=\{(\Re (x^*U\Sigma V^*x),\Im(x^*U\Sigma V^*x))\mid\|\Sigma V^*x\|_2=1\}.$$
Let $y=V\Sigma^{-1}x$, then $\Sigma=\mathrm{diag}\{\sigma_1,\ldots,\sigma_n\}$, then $$S_2:=\{(\Re (y^*\Sigma^{-1}V^*Uy),\Im(y^*\Sigma^{-1}V^*Uy))\mid\|y\|_2=1\},$$ so we can plot $S_2$ for the matrix $\Sigma^{-1}V^*U$.
I have tried to plot $S_1$ and $S_2$ for different random $A$, and found that $S_1\subset S_2$, which is clearly not equal to $S$, in set $S$ we have more constraints so I expect to get $S\subset S_1$. Anyone please can help me to fix my mistake?