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 .