# Definition of Markov matrix

Put content here**Definition:** A **Markov matrix** (or **stochastic matrix**) is a square matrix used to describe transitions in a Markov chain. There are two types:
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- **Right stochastic**: each row sums to 1, entries are non-negative. $P_{ij}$ = probability of transitioning from state $i$ to state $j$.
- **Left stochastic**: each column sums to 1, entries are non-negative.
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$$P = \begin{pmatrix} 0.7 & 0.2 & 0.1 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.3 & 0.5 \end{pmatrix}$$
(Each row sums to 1.)
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**Properties:**
- 1 is always an eigenvalue
- All eigenvalues satisfy $|\lambda| \leq 1$
- For a regular Markov chain, $P^n$ converges to a rank-1 matrix as $n \to \infty$
- The steady-state vector $\pi$ satisfies $\pi P = \pi$ (left eigenvector with eigenvalue 1)
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Markov matrices are fundamental in probability theory, statistics, and applications like PageRank.

# Parents

* Particular types of matrices