# 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

1. $$\Gamma$$ is skew symmetric matrix i.e, $$\Gamma (u,v) = -\Gamma (v,u) \quad ,\forall \, u,v \in \Omega$$

2. Row sum of $$\Gamma$$ is zero for every row i.e, $$\sum_v \Gamma (u,v) = 0 \quad ,\forall \, u \in \Omega$$

3. Third property is, $$\Gamma(u,v) > -\pi (v)Q(v,u) \quad ,\forall \, u,v \in \Omega$$

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 $$\pi$$, $$\sum_u \pi(u) P(u,v) = \pi(v) \quad ,\forall \, v \in \Omega$$

• Changing third property to $\Gamma \geq \epsilon + \dots$ for some small $\epsilon$, you can use LP for this. – Eugene Aug 4 at 19:42