I am trying to solve the following homogeneous linear equation:
$$SB = \tilde{B}S$$
where
$$ B=\left( \begin{array}{ccc} \lambda & -\lambda & 0 \\ 0 & \lambda +\mu & -\lambda \\ 0 & 0 & \mu \\ \end{array} \right) $$
$$ \tilde{B} = B_{diag} = \left( \begin{array}{ccc} \lambda & 0 & 0 \\ 0 & \lambda +\mu & 0 \\ 0 & 0 & \mu \\ \end{array} \right) $$
for some $S$ where $S \epsilon'$ = 1 (that is the row sums of the $S$ matrix sum to 1). Here, $S$ would be a $3 \times 3$ matrix.
My attempt:
I used the Mathematica command $LinearSolve[S.B, Bdiag.S]$ and then used the solution to that, but to no avail. Any help on this would be appreciated.
Here is the Mathematica code from the equations below - there was no solution.
$Solve\left[\left\{\mu s_{12}-\lambda s_{11}=0, s_{13} (\mu -\lambda )-\lambda s_{12}=0, \mu s_{21}=0,\lambda s_{21}=0, \lambda \left(s_{22}+s_{23}\right)=0, s_{31} (\lambda -\mu )=0,\lambda \left(s_{32} s_{31}\right)=0, \lambda s_{32}=0,s_{11}+s_{12}+s_{13}=1,s_{21}+s_{22}+s_{23}=1,s_{31}+s_{32}+s_{33}=1\right\},\left\{s_{11},s_{12},s_{13},s_{21},s_{22},s_{23},s_{31},s_{32},s_{33}\right\}\right]$
In order for this system to have a solution the following set of equations must be satisfied:
\begin{align} \mu s_{12} - \lambda s_{11} &= 0\\ (\mu - \lambda)s_{13} - \lambda s_{12} &= 0\\ \mu s_{21} &= 0\\ \lambda s_{21} &= 0\\ \lambda (s_{22} + s_{23}) &= 0\\ (\lambda - \mu)s_{31} &= 0\\ \lambda(s_{32}-s_{31}) &= 0\\ \lambda s_{32} &= 0\\ s_{11} + s_{12} + s_{13} &= 1\\ s_{21} + s_{22} + s_{23} &= 1\\ s_{31} + s_{32} + s_{33} &= 1 \end{align}
If both $\lambda \neq 0$ and $\mu \neq 0$, then it is not possible to satisfy these equations since you would have $s_{21} = 0$ and $s_{22} + s_{23} = 0$, in which case $s_{21} + s_{22} + s_{23}$ clearly cannot sum to $1$.