I stumbled upon this system with constants $a_{i,j}>0$ that I want to solve for $x,y,z \in\mathbb{R}$:
\begin{align} a_{2,1}y+a_{3,1}z=& x(y+z) \\ a_{1,2}x+a_{3,2}z=& y(x+z) \\ a_{1,3}x+a_{2,3}y=& z(x+y) \end{align}
I would appreciate help on the following questions:
- To which class of equations does it belong?
- Is the solution unique (except the trivial solution (0,0,0))?
- How do I solve it in theory resp. approximate the solution effectively?
- Can 1-3 be generalized to n instead of 3 dimensions?
Edit:
I transformed the system to a matrix equation: Let \begin{equation} A= \begin{pmatrix} 0 & a_{1,2} & a_{1,3} \\ a_{2,1} & 0 & a_{2,3} \\ a_{3,1} & a_{3,2} & 0 \end{pmatrix}, \end{equation}
\begin{equation} B= \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \end{equation} and $\pi=(x,y,z)$. Then the system is just \begin{equation} \langle\pi,\pi B\rangle=\pi A \end{equation}, where $\langle.,.\rangle$ is the elementwise multiplication of two vectors. I found that with an arbitrary starting vector $\pi_0$, the following sequence should converge to the unique solution $\pi$: \begin{equation} \pi_{i+1}:=\frac{\pi_{i}A}{\pi_{i}B}, \end{equation} where the fraction is again elementwise.
Can anyone help me prove that? I could imagine a connection to the stationary distribution of irreducible Markov Chains, but I am not sure.
Also thankful for references to literature or other posts. Thanks!
this is a good question !
if you transform the matrix to inner product form, this question seems much easier:
\begin{equation} \langle\pi,\pi B\rangle=\pi A \end{equation}
you just need to calculate this equation immediately !
if this matrix is symmetry, then $xyz=a_{1}a_{2}a_{3}$
in the general case ,it seems like :
$xyz=a_{2,1}a_{1,3}a_{3,2}-a_{1,2}a_{2,3}a_{3,1}$