# Applications to Markov chains

Put content here.## Application to Markov Chains
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A **Markov chain** is a stochastic model describing a sequence of possible events where the probability of each state depends only on the current state (the Markov property). Linear algebra provides the framework for analyzing and computing properties of Markov chains.
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### Transition Matrix
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For a Markov chain with n states, the transition probabilities are encoded in an n × n **stochastic matrix** P, where Pᵢⱼ is the probability of transitioning from state i to state j. Each row sums to 1.
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### State Evolution
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If **x₀** is the initial state probability vector, the state after k steps is:
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**xₖ = Pᵏx₀**
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### Steady-State Distribution
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If the Markov chain is regular (some power of P has all positive entries), the system approaches a unique **steady-state vector** **q** such that:
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**Pq = q**
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This means **q** is an eigenvector of P corresponding to eigenvalue λ = 1.
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### Example
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Consider a simple weather model with states "Sunny" and "Rainy":
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```
P = [[0.9, 0.1],
     [0.5, 0.5]]
```
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The steady-state vector solves (P − I)q = 0 with q₁ + q₂ = 1, giving **q = [5/6, 1/6]ᵀ**. In the long run, 83.3% of days are sunny.
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### Applications
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Markov chains are used in:
- **PageRank** algorithm (Google search)
- Queueing theory and operations research
- Population genetics
- Financial modeling
- Natural language processing

# Parents

* Applications