In mathematics, especially in probability and combinatorics, a doubly stochastic matrix
(also called bistochastic matrix) is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1,[1] i.e.,
Thus, a doubly stochastic matrix is both left stochastic and right stochastic.[1][2]
Indeed, any matrix that is both left and right stochastic must be square: if every row sums to 1 then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal.[1]
^ abcGagniuc, Paul A. (2017). Markov Chains: From Theory to Implementation and Experimentation. USA, NJ: John Wiley & Sons. pp. 9–11. ISBN 978-1-119-38755-8.
^Marshal, Olkin (1979). Inequalities: Theory of Majorization and Its Applications. pp. 8. ISBN 978-0-12-473750-1.
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for existence is similar with Birkhoff–von Neumann theorem for doublystochasticmatrix. Here is an example that illustrates the existence and non-uniqueness...
Marvin; Newman, Morris (1959). "On the minimum of the permanent of a doublystochasticmatrix". Duke Mathematical Journal. 26. doi:10.1215/S0012-7094-59-02606-7...
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