Solution to $Mx-x^2=0$ where $x^2$ is the square of the elements of vector $x$

Question

I have been trying to find the solutions for

$$Mx=x\circ x$$ where $\circ$ is the element wise product.

One solution is $x=0$. But there is another solution $x\neq 0$, if $M$ has a real positive eigenvalue. ( $M$ is invertible, and quasipositive).

Any hints or references where to find more on this?

It is similar to the logistic equation, but multivariable with cross terms.

Cheers

Answer 1

Denote by $\text{diag}(x)$ the diagonal matrix whose elements are the elements of $x$, then you have: $$ (M-\text{diag}(x))x=0 $$ This implies: $$ x^T(M-\text{diag}(x))^T(M-\text{diag}(x))x=0 $$

Which is a polynomial equation on the entries of $x$, hence you should be able to find a solution with numerical methods, but I doubt you would find an analytical one.