Doubly stochastic matrix

Remarks

Clause 1 says that $A$ is a nonnegative matrix. With $e = (1, \ldots, 1)^T$, clauses 2 and 3 say $Ae = e$ and $e^T A = e^T$. Example: $A = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$; all entries are $\frac{1}{2} \ge 0$, and each row and each column sums to $1$. Every permutation matrix is doubly stochastic. Non-example: $\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$ satisfies clauses 1 and 2, but its columns sum to $2$ and $0$, so clause 3 fails. The set of doubly stochastic $n \times n$ matrices is the Birkhoff polytope.