Transition Matrices for Markov Chains

Problem

Transition matrix

Step 1: Define the Transition Matrix

A transition matrix describes the probabilities of moving between states in a system. Each entry $P_{ij}$ represents the probability of moving from state $i$ to state $j$.

Step 2: Check Row Sums

Every row of a transition matrix must sum to exactly $1$, since the total probability of moving from any given state to all possible next states is $1$.

Step 3: Find the Next State Distribution

To find the next state distribution, multiply the current state vector by the transition matrix:

$$v_1 = v_0 P$$