# Intersection of the sets of generalized doubly stochastic matrices and of orthogonal matrices

The definition of a doubly stochastic matrix can be found here. We say a square matrix $$A$$ is a generalized doubly stochastic matrix if the sums of each rows and columns of $$A$$ all equal $$1$$, but $$A$$ doesn't have to be non-negative.

An interesting fact (which is also easy to prove) about doubly stochastic matrices is: if $$A$$ is doubly stochastic and orthogonal, then $$A$$ is actually a permutation matrix.

What is the intersection set for a generalized doubly stochastic matrix set and orthogonal matrix set? More specifically, can any one give me an example of an $$N \times N$$ matrix $$A$$, which satisfy the following constraints:

• $$AA^T=I$$

• $$A 1=1$$

• $$A^T 1=1$$

• there exists at least one entry $$A_{i,j}$$, satisfying $$A_{i,j}<0$$

Sure, just take any solution to $x+y+z=x^2+y^2+z^2=1$ and form the matrix $$\left(\begin{array}{ccc} x&y&z\\y&z&x\\z&x&y\end{array}\right).$$
For instance, take $x = -\frac13$ and $y = z = \frac23$. More generally, choose the roots of the cubic $x^3 - x^2 + c$ for some small positive real $c$.