Suppose the elements of are . In other words the element in row , column is . Suppose also that the elements of are , and the elements of are
Now for each is formed when the elements of row of matrix are combined (by multiplication and addition) with the elements of . In other words:
And 's elements are a re-arrangement of 's elements. So for each for some . Therefore, from the above equation:
So each row of contains a single , and zeros. Moreover, each is equal to a different , so no column of contains more than a single . So is bistochastic, with elements and .