Recently, I read a book on cluster algebra and come across a problem that could finally be reduced to a problem of solving a system of nonlinear equations.
The question is:
Give $b_{12},b_{13} , b_{14},b_{23},b_{24},b_{34}\in \mathbb{C}$ satisfying $b_{12}b_{34}+b_{14}b_{23}-b_{13}b_{24}=0$.
Solve the system of nonlinear equations:
$\begin{cases} a_{11}a_{22}-a_{12}a_{21}=b_{12}\\ a_{11}a_{23}-a_{13}a_{21}=b_{13}\\ a_{11}a_{24}-a_{14}a_{21}=b_{14}\\ a_{12}a_{23}-a_{13}a_{22}=b_{23}\\ a_{12}a_{24}-a_{14}a_{22}=b_{24}\\ a_{13}a_{24}-a_{14}a_{23}=b_{34}\\ \end{cases}$
I want to solve the above system of nonlinear equations, but I don't know anything about systems of nonlinear equations.
Is there a solution to the system of equations? If so, what is the relationship between the solutions?
If anyone can give me some reference books or methods, I would be very grateful!
Suppose that $b_{12}$ is nonzero.
If you assume that $$\left[\begin{array}{cc} a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right]$$ is $b_{12}$ times an identity matrix, then you will find that you get linear equations to solve for the other entries.
Thinking of the entries $a_{ij}$ as a $2\times 4$ matrix, you can multiply it on the left by a two-by-two matrix with determinant 1, and this will give you all the other possible solutions.
If $b_{12}$ is zero, you can use a somewhat similar approach, first assuming that $a_{21}$ and $a_{22}$ are zero, and then, again, getting general solutions by multiplying by an invertible two-by-two matrix.