Consider the Markov chain with transition matrix
\[ P = \left( \begin{bmatrix} 1/2 & 1/2 \\ 1/4 & 3/4 \end{bmatrix}\right)\]
Find the fundamental matrix Z for this chain. Compute the mean first passage matrix using Z.
Solve the equation to calculate the fundamental matrix Z for the given Markov chain with transition matrix P,
where P - the transition matrix and I - the identity matrx
(P-I)Z = 0
\[ (P-I) = \left( \begin{bmatrix} 1/2-1 & 1/2 \\ 1/4 & 3/4-1 \end{bmatrix}\right)\]
\[ (P-I) = \left( \begin{bmatrix} -1/2 & 1/2 \\ 1/4 & -1/4 \end{bmatrix}\right) \left( \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix}\right) = \left( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\right)\]
\[-1/2z_{11} + 1/2z_{21} = 0\] \[1/4z_{11} - 1/4z_{12} = 0\] \[-1/2z_{12} + 1/2z_{22} = 0\] \[1/4z_{21} - 1/4z_{22} = 0\]
After solving the equation
\[z_{11} = z_{21}\] \[z_{11} = z_{12}\] \[z_{12} = z_{22}\] \[z_{21} = z_{22}\]
Sum of each row in the fundamental matrix Z should be equal to 1.
Now normalize so the sum of rows=1
Assume \[z_{11} = 1\]
\[Z = \left( \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\right)\]
To find first passage matrix N using Z, apply the formula:
\[N= Z-I\]
\[N= \left( \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\right)-\left( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\right)= \left( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\right)\]
Mean First Passage Matrix N:
\[N= \left( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\right)\]
Fundamental Matrix Z for the given Markov Chain:
\[Z = \left( \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\right)\]