Denote $x=[x_{1},x_{2},\cdots,x_{n}]\in\mathbb{R}^{nd}$, where $x_{i}\in\mathbb{R}^{d}$ for $i=1,2\cdots,n$. Suppose a matrix $A\in\mathbb{R}^{k\times n}$,$B=A\otimes I_{d}\in\mathbb{R}^{kd\times nd}$ where $I_{d}$ is the unity matrix. And $x$ solves the linear equation: $Ax=0$ subject to $||x_{1}||=||x_{2}||=\cdots=||x_{n}||>0$, where $||\cdot||$ is the standard 2-norm on Euclidean space. One can observe that if $x^{\ast}$ is a solution satisfying the norm constraints, for any constant $c\neq 0$, and orthogonal matrix $R\in\mathcal{O}(d)$, $cRx^{\ast}$ is also a solution satisfying constraints. This means if there exist feasible solutions, the solution space is a manifold obtained as the union of orbits of some specific solutions. The solution manifold may be disconnected. I wonder whether there exist some theorical results about the number of connect components? Perhaps some interesting topological results exist. Furthermore, to characterize the solution manifold, is there some numerical efficient way in time complexity? Up to now, I can only come up with semi-definite programming method. Can this method ensure find all points on disconnect components?
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