Doubly stochastic matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially in <a href="/facts/Probability/fgYvMhwb">probability</a> and <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, a doubly stochastic matrix 
(also called bistochastic matrix) is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> 
 
 
 
 X
 =
 (
 
 x
 
 i
 j
 
 
 )
 
 
 {\displaystyle X=(x_{ij})}
 
 of nonnegative <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, each of whose rows and columns sums to 1, i.e.,

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

Thus, a doubly stochastic matrix is both left <a href="/facts/Stochastic_matrix/2cRWAqC3">stochastic</a> and right stochastic.
Indeed, any matrix that is both left and right stochastic must be <a href="/facts/Square_matrix/5TP9mNNn">square</a>: 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.

Doubly stochastic matrix open-in-new

Doubly stochastic matrix