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Mathematics-Online course: Linear Algebra - Matrices - Special Matrices

Stochastic Matrices

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A given matrix $ P\in\mathbb{R}^{n\times n}$ is called stochastic if $ p_{i,j}\ge 0$ and

$\displaystyle \sum_{j=1}^n p_{i,j} = 1,\quad i=1,\ldots,n\, . $

Stochastic matrices describe the change of probabilities $ x_i$ according to

$\displaystyle x'_j = \sum_i x_i p_{i,j}\, . $

(Authors: App/Burkhardt/Höllig)
Because of the properties of $ P$ vector $ x$ is a vector of probabilities, that is,

$\displaystyle x'_j \ge 0,\quad \sum_j x'_j = 1\, . $

Non-negativity is obvious and the invariance of the sum follows from

$\displaystyle \sum_j \sum_i x_i p_{i,j} = \sum_i x_i \underbrace{\sum_j p_{i,j}}_{=1}\, . $

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005