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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 $A_{i,j}<0$

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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 $c$.

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  • $\begingroup$ Thanks, Erick, This is a nice example. $\endgroup$
    – Benson
    Commented Jun 17, 2012 at 2:24

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