Given a pair of distributions $x,y\in(0,1]^{n\times 1}$, so that $1^Tx=1$ and $1^Ty=1$,
I want to build a matrix $C$ (change matrix) that satisfy **at least** the following properties:

i) $C$ is diagonal if and only if $x=y$

ii) $C1 = x$

iii) $C^T1 = y$

iv) $C$ has nonnegative entries

How to build a $C$ that satisfy i)-iv)?

If $\Lambda_x = diag(x)$ and $\Lambda_y = diag(y)$ conditions ii) and iii) can be also written as:

(1) $C\Lambda_y^{-1}y = x$

(2) $C^T\Lambda_x^{-1}x = y$

respectivelly. Replacing (2) in (1) results in:

(3) $C\Lambda_y^{-1}C^T\Lambda_x^{-1}x = x$

And replacing $x$ by $\Lambda_x1$ results the matricial Equation:

(4) $\(C\Lambda_y^{-1}C^T-\Lambda_x\)1 = 0$

or alternativelly (if 1 is replaced in 2),

(5) $\(C^T\Lambda_x^{-1}C-\Lambda_y\)1 = 0$