Question
Prove that, for all $0 \leq p, q \leq 1$ with $p+q>0$, the probability eigenvector of the transition matrix $T=\left(\begin{array}{cc}1-p & q \\ p & 1-q\end{array}\right)$ is $\mathbf{v}=\left(\frac{q}{p+q}, \frac{p}{p+q}\right)^T$.
Step 1
The transition matrix is given by \( T = \begin{pmatrix} 1-p & q \\ p & 1-q \end{pmatrix} \). We need to prove that the vector \( \mathbf{v} = \begin{pmatrix} \frac{q}{p+q} \\ \frac{p}{p+q} \end{pmatrix} \) is an eigenvector of \( T \). Show more…
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