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# Stochastic Matrix

A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize transitions for a finite Markov chain, Elements of the matrix must be real numbers in the closed interval [0, 1].

A completely independent type of stochastic matrix is defined as a square matrix with entries in a field such that the sum of elements in each column equals 1. There are two nonsingular stochastic matrices over (i.e., the integers mod 2),

There are six nonsingular stochastic matrices over ,

In fact, the set of all nonsingular stochastic matrices over a field forms a group under matrix multiplication. This group is called the stochastic group.

The following tables give the number of distinct stochastic matrices (and distinct nonsingular stochastic matrices) over for small .

 stochastic matrices over 2 1, 4, 64, 4096, ... 3 1, 9, 729, ... 4 1, 16, 4096, ...
 stochastic nonsingular matrices over 2 1, 2, 24, 1440, ... 3 1, 6, 450, ... 4 1, 12, 3108, ...

Doubly Stochastic Matrix, Horn's Theorem, Majorization, Markov Chain, Stochastic Group

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## References

Poole, D. G. "The Stochastic Group." Amer. Math. Monthly 102, 798-801, 1995.

## Referenced on Wolfram|Alpha

Stochastic Matrix

## Cite this as:

Weisstein, Eric W. "Stochastic Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StochasticMatrix.html