there are no non-square doubly stochastic matrices


Suppose A=(aij) is a n×m matrix with nonnegative entries such that

j=1maij = 1,i=1,,n, (1)
i=1naij = 1,j=1,,m. (2)

Then n=m.

This is seen by summing equation (1) over i=1,,n and equation (2) over j=1,,m. Then

i=1nj=1maij = n,
i=1nj=1maij = m,

and since the right hand sides coincide, it follows that n=m.

Title there are no non-square doubly stochastic matrices
Canonical name ThereAreNoNonsquareDoublyStochasticMatrices
Date of creation 2013-03-22 15:11:00
Last modified on 2013-03-22 15:11:00
Owner matte (1858)
Last modified by matte (1858)
Numerical id 8
Author matte (1858)
Entry type Result
Classification msc 15A51
Classification msc 60G99