Vorticity Matrix for Markov chain I have a markov chain with $Q(u,v)$ as transition probability matrix and $\pi(u)$ as stationary distribution defined on state space $\Omega$. The dimension of matrix $Q$ is $nxn$ and vector $\pi$ is $1xn$.  

I need to construct a vorticity matrix $\Gamma (u,v)$ of dimension $nxn$ which has below properties


$\Gamma$ is skew symmetric matrix i.e, $$\Gamma (u,v) = -\Gamma (v,u) \quad ,\forall \, u,v \in \Omega $$
Row sum of $\Gamma$ is zero for every row i.e, $$ \sum_v \Gamma (u,v) = 0 \quad ,\forall \, u \in \Omega $$
Third property is, $$\Gamma(u,v) &gt; -\pi (v)Q(v,u) \quad ,\forall \, u,v \in \Omega  $$  


My question is : How to construct vorticity matrix $\Gamma (u,v)$ which satisfies above three properties? I need to construct at least one such matrix. 

Is there any systematic way to build such matrices 

NOTE: Transition probability matrix  $P$, and stationary distribution $\pi$ has below properties

Row sum of $P$ is one for each row,
 $$\sum_v P(u,v)=1 \quad ,\forall \, u \in \Omega$$
$\pi$ is probability distribution hence,
 $$\sum_v \pi(v) = 1$$
Stationary distribution condition for $\pi$,
 $$\sum_u \pi(u) P(u,v)  = \pi(v) \quad ,\forall \, v \in \Omega $$
One can verify that the following matrix $\Gamma$ has the desired properties:
$\Gamma = [\pi]Q - Q^{\top}[\pi]$,
where $[\pi]$ is a diagonal matrix with diagonal elements being the stationary distribution. This construction relates closely to the reversibility of a Markov chain as far as I know.