A probability vector $\mathbf{u}$ is a vector where each element $u_i \geq 0$ and the sum of all elements equals 1, i.e., $\sum_i u_i = 1$. A transition matrix $T$ is a square matrix where each element $t_{ij} \geq 0$, and each column sums to 1, i.e., $\sum_i
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