A **doubly stochastic matrix** is a special type of square matrix that has non-negative entries and each row and each column sums to 1. In other words, for a matrix \( A \) of size \( n \times n \), the following conditions must hold: 1. \( a_{ij} \geq 0 \) for all \( i, j \) (all entries are non-negative).
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A square real matrix is doubly stochastic if its entries are nonnegative and every row and column sums to . Its positive-entry support satisfies the condition of the Hall marriage theorem: for a row subset , by summing column capacities. Consequently its support contains a permutation matrix.