13. (Sinkhorn) 设 $A$ 是一个方的正矩阵, 则存在对角元素为正数的两个对角矩阵 $D_1$ 和 $D_2$ 使得 $D_1AD_2$ 为双随机矩阵 (doubly stochastic matrix).

证明: 见 [R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist., 35 (1964), 876--879].