Doubly stochastic matrix

In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix ${\displaystyle X=(x_{ij})}$ of nonnegative real numbers, each of whose rows and columns sums to 1, i.e.,

${\displaystyle \sum _{i}x_{ij}=\sum _{j}x_{ij}=1,}$

Thus, a doubly stochastic matrix is both left stochastic and right stochastic.^[1]

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.