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The definition for doubly stochastic matrix can be found here. We say a square matrix $A$ is a Generalized doubly stochastic matrix if the sum of each rows and columns of $A$ all equals 1. But A doesn't have to be non-negative.

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

So my question is: 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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Thanks, Erick, This is a nice example. – Benson Jun 17 '12 at 2:24

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