I am trying to solve this set of linear equations:
$$\left( \begin{array}{ccc} s_{12} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \\ \end{array} \right)\left( \begin{array}{ccc} \lambda & -\lambda & 0 \\ 0 & \lambda +\mu & -\lambda \\ 0 & 0 & \mu \\ \end{array} \right)-\left( \begin{array}{ccc} \lambda & 0 & 0 \\ 0 & \lambda +\mu & 0 \\ 0 & 0 & \mu \\ \end{array} \right)\left( \begin{array}{ccc} s_{12} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \\ \end{array} \right) =\bf{0}$$
and I get no solution.
But, when I change it to
$$\left( \begin{array}{ccc} s_{12} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \\ \end{array} \right) \left( \begin{array}{ccc} \lambda _1 & -\lambda _1 & 0 \\ 0 & \lambda _2+\mu _1 & -\lambda _2 \\ 0 & 0 & \mu _2 \\ \end{array} \right)-\left( \begin{array}{ccc} s_{12} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \\ \end{array} \right) \left( \begin{array}{ccc} \lambda _1 & 0 & 0 \\ 0 & \lambda _2+\mu _1 & 0 \\ 0 & 0 & \mu _2 \\ \end{array} \right) = \bf{0}$$
I do get a set of solutions.
I can't seem to figure out why this happens. Any direction would be greatly appreciated.
For the second set, here's how I solved it. Just multiply the matrices out and we get...
$$\left( \begin{array}{ccc} 0 & s_{12} \left(-\lambda _1+\lambda _2+\mu _1\right)-s_{11} \lambda _1 & s_{13} \left(\mu _2-\lambda _1\right)-s_{12} \lambda _2 \\ s_{21} \left(\lambda _1-\lambda _2-\mu _1\right) & -s_{21} \lambda _1 & -s_{22} \lambda _2-s_{23} \left(\lambda _2+\mu _1-\mu _2\right) \\ s_{31} \left(\lambda _1-\mu _2\right) & s_{32} \left(\lambda _2+\mu _1-\mu _2\right)-s_{31} \lambda _1 & -s_{32} \lambda _2 \\ \end{array} \right)$$
and then solve this set of linear equations...
$$s_{12} \left(-\lambda _1+\lambda _2+\mu _1\right)-\lambda _1 s_{11}=0, \\s_{13} \left(\mu _2-\lambda _1\right)-\lambda _2 s_{12}=0, s_{21} \left(\lambda _1-\lambda _2-\mu _1\right)=0,\\ \lambda _1 \left(-s_{21}\right)=0,\\ \lambda _2 \left(-s_{22}\right)-s_{23} \left(\lambda _2+\mu _1-\mu _2\right)=0,\\ s_{31} \left(\lambda _1-\mu _2\right)=0, \\ s_{32} \left(\lambda _2+\mu _1-\mu _2\right)-\lambda _1 s_{31}=0, \\ \lambda _2 \left(-s_{32}\right)=0, \\ s_{11}+s_{12}+s_{13}=1, \\ s_{21}+s_{22}+s_{23}=1 \\ s_{31}+s_{32}+s_{33}=1$$
To get:
$$s_{11} = -\frac{\left(\lambda _1-\lambda _2-\mu _1\right) \left(\lambda _1-\mu _2\right)}{\lambda _1 \mu _1-\lambda _2 \mu _2-\mu _2 \mu _1},s_{12} = \frac{\lambda _1 \left(\lambda _1-\mu _2\right)}{\lambda _1 \mu _1-\lambda _2 \mu _2-\mu _2 \mu _1},s_{13} = \frac{\lambda _1 \lambda _2}{-\lambda _1 \mu _1+\lambda _2 \mu _2+\mu _2 \mu _1},s_{21} = 0,s_{22} = \frac{\lambda _2}{\mu _1-\mu _2}+1,s_{23} = -\frac{\lambda _2}{\mu _1-\mu _2},s_{31} = 0,s_{32} = 0,s_{33} = 1$$
There is no mystery. The two systems have no reason to be both solvable, they have nothing in common.
The first system has the form
$$S\Lambda-\Lambda'S=0$$ while the second is
$$S\Lambda-S\Lambda'=S(\Lambda-\Lambda')=0$$ where $\Lambda-\Lambda'$ has a single nonzero element (the solution is $s_{11}=s_{21}=s_{31}=0$ or $\lambda_1=0$).