Steady state vector calculator
This calculator is for calculating the steady-state of the Markov chain stochastic matrix. A very detailed step by step solution is provided. This matrix describes the transitions of a Markov chain. This matric is also called as probability matrix, transition matrix, etc
You can see a sample solution below. Enter your data to get the solution for your question
$$ \displaylines{---} $$
$$ \displaylines{ \mathbf{\color{Green}{Let's\;call\;All\;possible\;states\;as\;}}
\begin{bmatrix}
1 & 2 & \end{bmatrix}
\\ \\
\mathbf{\color{Green}{First\;we\;have\;to\;create\;Stochastic\;matrix}}
\\ \\
Stochastic\;matrix\;=\;P=
\begin{bmatrix}
* & 1 & 2 & \\ \\
1 & 0.5 & 0.5 & \\ \\
2 & 0.8 & 0.2 & \end{bmatrix}
\\ \\
\\ \\
\mathbf{\color{Green}{For\;steady\;state.\;We\;have\;to\;solve\;these\;equation}}
\\ \\
X*P=X
\\ \\
and\;
x_{1}+x_{2}
=1
\\ \\
Where\;X\;=\;
\begin{bmatrix}
x_{1} & x_{2} & \end{bmatrix}
\\ \\
P=
\begin{bmatrix}
0.5 & 0.5 & \\ \\
0.8 & 0.2 & \end{bmatrix}
\\ \\
\mathbf{\color{Green}{That\;is\;}}
\\ \\ \Rightarrow
\begin{bmatrix}
x_{1} & x_{2} & \end{bmatrix}
\begin{bmatrix}
0.5 & 0.5 & \\ \\
0.8 & 0.2 & \end{bmatrix}
=
\begin{bmatrix}
x_{1} & x_{2} & \end{bmatrix}
\\ \\
and\;
x_{1}+x_{2}
=1
\\ \\
\mathbf{\color{Green}{Simplifying\;that\;will\;give}}
\\ \\
x_{1}*(0.5)+x_{2}*(0.8)=x_{1}
\\ \\
x_{1}*(0.5)+x_{2}*(0.2)=x_{2}
\\ \\
x_{1}+x_{2}
=1
\\ \\
\mathbf{\color{Green}{Simplifying\;again\;will\;give}}
\\ \\
x_{1}*(-0.5)+x_{2}*(0.8)=0
\\ \\
x_{1}*(0.5)+x_{2}*(-0.8)=0
\\ \\
x_{1}+x_{2}
=1
\\ \\
\mathbf{\color{Green}{Solving\;those\;will\;give\;below\;result...}}
\\ \\
\begin{bmatrix}
x_{1} & x_{2} & \end{bmatrix}
=
\begin{bmatrix}
0.615385 & 0.384615 & \end{bmatrix}
} $$