P=\begin{pmatrix} 0.25 & 0.25 & 0.50 \\ 0.10 & 0.30 & 0.60\\ 0.05 & 0.15 & 0.80\\ \end{pmatrix} I know that $\vec{u}= P\vec{u}$ and we are trying to get:
$(I-P)\vec{u}=\vec{0}$
I know that $(I-P)$= \begin{pmatrix} 0.75 & -0.25 &- 0.50 \\ -0.10 & 0.70 & -0.60\\ -0.05 & -0.15 & 0.20\\ \end{pmatrix}
So know I need to find $\vec{u}$ such that :
$ \begin{pmatrix} 0.75 & -0.25 &- 0.50 \\ -0.10 & 0.70 & -0.60\\ -0.05 & -0.15 & 0.20\\ \end{pmatrix} * \begin{pmatrix} x_1 \\ x_2\\ x_3\\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0\\0\\ \end{pmatrix} $
Can I make $(I-P)$ into this: \begin{pmatrix} 0.75 & -0.25 &- 0.50 &0 \\ -0.10 & 0.70 & -0.60&0\\ 1 & 1 & 1 &1\\ \end{pmatrix} and apply the gauss jordan ? I just need guidance on how to calculate the $\vec{u}$.
Thank you,
If I do the gauss-jordan on $(I-P)$ I get $\vec{u}$ = \begin{pmatrix} 1/3 \\ 1/3\\ 1/3\\ \end{pmatrix} then $(I-P)\vec{u}=\vec{0}$ so I do get 0 as a result, but I don't know if the solution is correct.
Assume that the transition matrix shows the percent of customers of $A,B,C$ visiting next time. $$P=\begin{pmatrix} P_{AA} & P_{AB} & P_{AC} \\ P_{BA} & P_{BB} & P_{BC}\\ P_{CA} & P_{CB} & P_{CC}\\ \end{pmatrix}=\begin{pmatrix} 0.25 & 0.25 & 0.50 \\ 0.10 & 0.30 & 0.60\\ 0.05 & 0.15 & 0.80\\ \end{pmatrix}.$$ Now you want to find the steady state: $$x_n=x_{n-1}P \Rightarrow x=xP \Rightarrow x(I-P)=0 \Rightarrow \\ \begin{pmatrix}x_1&x_2&x_3\end{pmatrix}\begin{pmatrix} \ \ \ 0.75 & -0.25 & -0.50 \\ -0.10 & \ \ \ 0.70 & -0.60\\ -0.05 & -0.15 & \ \ \ 0.20\\ \end{pmatrix}=0\\ \begin{cases} 0.75x_1-0.1x_2-0.05x_3=0\\ -0.25x_1+0.7x_2-0.15x_3=0\\ x_1+x_2+x_3=1\end{cases} \Rightarrow \\ \begin{pmatrix}x_1&x_2&x_3\end{pmatrix}=\begin{pmatrix}\frac2{27}&\frac5{27}&\frac{20}{27}\end{pmatrix}.$$