Let
A
A
be a nonnegative
m
×
n
m \times n
matrix and let
r
=
(
r
1
,
⋯
,
r
m
)
r = ({r_1}, \cdots ,{r_m})
and
c
=
(
c
1
,
⋯
,
c
n
)
c = ({c_1}, \cdots ,{c_n})
be positive vectors such that
Σ
i
=
1
m
r
i
=
Σ
j
=
1
n
c
j
\Sigma _{i = 1}^m{r_i} = \Sigma _{j = 1}^n{c_j}
. It is well known that if there exists a nonnegative
m
×
n
m \times n
matrix
B
B
with the same zero pattern as
A
A
having the
i
i
th row sum
r
i
{r_i}
and
j
j
th column sum
c
j
{c_j}
, there exist diagonal matrices
D
1
{D_1}
and
D
2
{D_2}
with positive main diagonals such that
D
1
A
D
2
{D_1}A{D_2}
has
i
i
th row sum
r
i
{r_i}
and
j
j
th column sum
c
j
{c_j}
. However the known proofs are at best cumbersome. It is shown here that this result can be obtained by considering the minimum of a certain real-valued function of
n
n
positive variables.