Stochastic matrix

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2020 Mathematics Subject Classification: Primary: 15B51 Secondary: 60J10 [MSN][ZBL]

A stochastic matrix is a square (possibly infinite) matrix $P=[p_{ij}]$ with non-negative elements, for which $$ \sum_j p_{ij} = 1 \quad \text{for all $i$.} $$ The set of all stochastic matrices of order $n$ is the convex hull of the set of $n^n$ stochastic matrices consisting of zeros and ones. Any stochastic matrix $P$ can be considered as the matrix of transition probabilities of a discrete Markov chain $\xi^P(t)$.

The absolute values of the eigenvalues of stochastic matrices do not exceed 1; 1 is an eigenvalue of any stochastic matrix. If a stochastic matrix $P$ is indecomposable (the Markov chain $\xi^P(t)$ has one class of positive states), then 1 is a simple eigenvalue of $P$ (i.e. it has multiplicity 1); in general, the multiplicity of the eigenvalue 1 coincides with the number of classes of positive states of the Markov chain $\xi^P(t)$. If a stochastic matrix is indecomposable and if the class of positive states of the Markov chain has period $d$, then the set of all eigenvalues of $P$, as a set of points in the complex plane, is mapped onto itself by rotation through an angle $2\pi/d$. When $d=1$, the stochastic matrix $P$ and the Markov chain $\xi^P(t)$ are called aperiodic.

The left eigenvectors $\pi = \{\pi_j\}$ of $P$ of finite order, corresponding to the eigenvalue 1: \begin{equation} \label{eq1} \pi_j = \sum_i \pi_i p_{ij} \quad \text{for all}\ j\,, \end{equation} and satisfying the conditions $\pi_j \geq 0$, $\sum_j\pi_j = 1$, define the stationary distributions of the Markov chain $\xi^P(t)$; in the case of an indecomposable matrix $P$, the stationary distribution is unique.

If $P$ is an indecomposable aperiodic stochastic matrix of finite order, then the following limit exists: \begin{equation} \label{eq2} \lim_{n\rightarrow\infty} P^n = \Pi, \end{equation} where $\Pi$ is the matrix all rows of which coincide with the vector $\pi$ (see also Markov chain, ergodic; for infinite stochastic matrices $P$, the system of equations \ref{eq1} may have no non-zero non-negative solutions that satisfy the condition $\sum_j \pi_j < \infty$; in this case $\Pi$ is the zero matrix). The rate of convergence in \ref{eq2} can be estimated by a geometric progression with any exponent $\rho$ that has absolute value greater than the absolute values of all the eigenvalues of $P$ other than 1.

If $P = [p_{ij}]$ is a stochastic matrix of order $n$, then any of its eigenvalues $\lambda$ satisfies the inequality (see [MM]): $$ \left| \lambda - \omega \right| \leq 1-\omega, \quad \text{where $\omega = \min_{1 \leq i \leq n} p_{ii}.$} $$ The union $M_n$ of the sets of eigenvalues of all stochastic matrices of order $n$ has been described (see [Ka]).

A stochastic matrix $P=[p_{ij}]$ that satisfies the extra condition $$ \sum_i p_{ij} = 1 \quad \text{for all $j$} $$ is called a doubly-stochastic matrix. The set of doubly-stochastic matrices of order $n$ is the convex hull of the set of $n!$ permutation matrices of order $n$ (i.e. doubly-stochastic matrices consisting of zeros and ones). A finite Markov chain $\xi^P(t)$ with a doubly-stochastic matrix $P$ has the uniform stationary distribution.


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Given a real $n\times n$ matrix $A$ with non-negative entries, the question arises whether there are invertible positive diagonal matrices $D_1$ and $D_2$ such that $D_1AD_2$ is a doubly-stochastic matrix, and to what extent the $D_1$ and $D_2$ are unique. Such theorems are known as $DAD$-theorems. They are of interest in telecommunications and statistics, [C], [F], [Kr].

A matrix $A$ is fully decomposable if there do not exist permutation matrices $P$ and $Q$ such that $$ PAQ = \left[ \begin{array}{cc} A_1 & 0 \\ B & A_2 \end{array} \right]. $$ A $1 \times 1$ matrix is fully indecomposable if it is non-zero.

Then for a non-negative square matrix $A$ there exist positive diagonal matrices $D_1$ and $D_2$ such that $D_1AD_2$ is doubly stochastic if and only if there exist permutation matrices $P$ and $Q$ such that $PAQ$ is a direct sum of fully indecomposable matrices [SK], [BPS].


[SK] R. Sinkhorn, P. Knopp, "Concerning nonnegative matrices and doubly stochastic matrices" Pacific J. Math. , 21 (1967) pp. 343–348 MR0210731 Zbl 0152.01403
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[F] S. Fienberg, "An iterative procedure for estimation in contingency tables" Ann. Math. Stat. , 41 (1970) pp. 907–917 MR0266394 Zbl 0198.23401
[Kr] R.S. Krupp, "Properties of Kruithof's projection method" Bell Systems Techn. J. , 58 (1979) pp. 517–538
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[S] E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) MR2209438 Zbl 0471.60001
How to Cite This Entry:
Stochastic matrix. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article