The relative gain array of an 
invertible matrix 
is the matrix
 |
(1)
|
where 
denotes the Hadamard matrix product and
denotes the transpose. The entries are 
. The transpose
is used even for complex entries, so no complex
conjugates are taken (Johnson and Shapiro 1986).
Every row and column of 
sums to 1, since these sums are entries on the diagonal
of 
and 
,
respectively. A real relative gain array is therefore
doubly stochastic iff
it is entrywise nonnegative. It need not be nonnegative
or invertible. Its matrix
inverse, when it exists, is the inverse
relative gain array.
The relative gain array is unchanged by multiplying individual rows and columns by nonzero scalars. More precisely, for invertible
matrices 
,
that are diagonal, and permutation
matrices 
,
,
 |  |  |
(2)
|
 |  |  |
(3)
|
For a complex Hadamard matrix 
of order 
,
 |
(4)
|
where 
is the unit matrix with every entry equal to 1. For
real matrices, the equation
has a solution for 
,
,
and every 
,
but has no solution for 
(Zhang et al. 1999, Trott 2006). For 
, one construction is
 |
(5)
|
where 
is the identity matrix and 
is either real root of 
. This quadratic
equation has polynomial discriminant 
, which is nonnegative
for 
.
To see the obstruction for 
, note that every entry of 
must be nonzero. Scaling then
makes the first row and column of 
equal to 1. Writing the second row as 
, the off-diagonal entries of 
give 
and 
. Thus 
and 
, so 
and 
would be real roots of 
, which has negative polynomial
discriminant. In contrast, a Fourier matrix
gives a complex solution in every order.
For every 
and positive integer 
, real solutions also exist
for
 |
(6)
|
where 
denotes 
-fold
composition, with every matrix
to which 
is applied invertible (Zhang et al. 1999,
Chen 2001).
." SIAM J. Matrix Anal. Appl. 22,
965-970, 2001. 
."
SIAM J. Algebraic Discrete Methods 7, 627-644, 1986. 
." SIAM J. Matrix Anal. Appl. 21,
642-645, 1999.