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For a given $A\in\mathbb{C}^{n\times n}$ and $a\in\mathbb{C}$, $|a|\leq1$ , assume that $x^*Ax=a$ holds for some unit vector $x\in\mathbb{C}^{n}$.
I want to find that $x$.

I couldn't solve it even for unitarily diagonalizable case: $A=U\Sigma U^*$. In that case we end up with two equations:

$$\frac{\sigma_1|y_1|^2}{a}+\frac{\sigma_2|y_2|^2}{a}+\cdots+\frac{\sigma_2|y_2|^2}{a}=1,$$ $$|y_1|^2+|y_2|^2+\cdots+|y_n|^2=1,$$

where $y=U^*x$ and $\sigma_i's$ are singular values of $A$. The two equations are ellipsoid and sphere equations, and there might be more than one $y$ that satisfy them. However, I need only one solution. Any comment on general case would be appreciated.

Lee
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    If $A$ is unitarily diagonalizable (equivalently if $A^A = AA^$), then $x^*Ax = a$ will have a solution if and only if $a$ lies in the convex hull of the eigenvalues of $A$. If you want an algorithm that produces a suitable $x$ given an arbitrary feasible $a$, then something can be done with the help of linear programming. See this post for details. – Ben Grossmann Jun 21 '21 at 04:17
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    If $A$ is unitarily diagonalizable with real eigenvalues (equivalently $A = A^*$), then it suffices to take $x$ to be a convex combination of the eigenvectors associated with the smallest and largest eigenvalues of $A$. – Ben Grossmann Jun 21 '21 at 04:18
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    You can very quickly see that this is not possible for arbitrary $a$ since the unit sphere is compact so the image of the function $x\mapsto x^*Ax$ is compact. – Eric Wofsey Jun 21 '21 at 04:30
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    The easiest way to see that this is not possible without further assumptions is to notice that it fails miserably for $n=1$. E.g., if $A=17$ and $a=42$, how can you find $x$ such that $x^*17x=42$ and $|x|_2=1$? – Gerry Myerson Jun 21 '21 at 06:22
  • OK, you've edited the question – have you tested what happens for $n=1$, $a=42$? I don't know what all those $\sigma_i$ are supposed to be, or what equations they might satisfy. – Gerry Myerson Jun 22 '21 at 12:59
  • @GerryMyerson yes, I think I got your point. Even if we maximize $x^*17x$ over unit vector $x$, we will get 17. $\sigma_i's$ are singular values of $A$ – Lee Jun 23 '21 at 01:43

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