Doubly stochastic matrix
Topic of Discrete-time Markov chains, part of Stochastic processes
Doubly stochastic matrix
| Type | Topic |
| Taught in | Discrete-time Markov chains |
| Part | Stochastic processes |
| Questions using it | 2 |
A transition matrix whose columns, as well as its rows, sum to 1. On m states the uniform distribution (1/m,\ldots,1/m) is then stationary. The condition is sufficient for a uniform stationary law, not necessary for a stationary law to exist.
In Discrete-time Markov chains
If P is doubly stochastic, every column sums to 1 as well as every row. On a finite set with m states the uniform row (1/m,\ldots,1/m) then satisfies
(\pi P)_j=\sum_{i=1}^m \frac{1}{m} p_{ij}=\frac{1}{m},
because column j sums to 1. Uniformity is stationary. Doubly stochastic is a sufficient condition, not a necessary one: the weather chain has a stationary distribution and its columns do not sum to 1.
Used in
- Taught in Discrete-time Markov chains
- Also used in Classes, recurrence, and reversibility
- Questions that use it (2) — A deterministic three-cycle (answer) · A doubly stochastic matrix
- All 3 questions on Discrete-time Markov chains
See also
Transition matrix · Uniform distribution · Discrete-time Markov chain · Stationary distribution · Hitting probabilities · Spectral view for two states · Markov assumption as a chosen state · Stationarity and the law from one start