Solution to a simple system of quadratic equations

Question

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations:

$$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$

for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial solution at $x=0$ but I am looking for others. Any help would be much appreciated.

Answer 1

Let $B\in\mathbb{R}^{n\times n}$ a matrix with entries $B_{ij}$. Let $x\in\mathbb{R}^n$ such that $x=(x_1,\ldots,x_n)$, define $y=(x_1^2,\ldots,x_n^2)$ and note that your system is equivalent to the system $B^Tx = y$. This is a linear system and there is a lot about this in the literature.