Solve Matrix Equation $\sqrt{x^TAx} - x^\top\theta = 0$

Question

let $x,\theta \in \mathbb{R}^n$, $A\in \mathbb{R}^{n\times n}$ Positive semi-definite and symmetrtic. How to solve (for $x \neq 0$) the following equation?

$$\sqrt{x^TAx} - x^\top\theta = 0$$

Answer 1

The equation can have an infinity of solutions. Try a simple trivial case. Choose $n=2$ and $A=I_2$. Then the general form of $x^T$ is $(x_1\ x_2)$. So the equation becomes $$\sqrt{x_1^2+x_2^2}-x_1\theta_1-x_2\theta_2=0$$ Moving the last two terms from the left to the right, and squaring the equation, you get: $$x_1^2+x_2^2=x_1^2\theta_1^2+x_2^2\theta_2^2+2x_1x_2\theta_1\theta_2$$ This is the equation of a conic (degenerate). In this case any point on the conic is a solution. Simpler way to see that, if $x$ is a solution, $ax$ is a solution as well, with $a>0$,