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 $c$.