Confusion related to gaussian loglikelihood maximization problem

I have this confusion related to gaussian log likelihood problem. I was reading this paper

The objective is to maximize the gaussian log likelihood given some samples from a multivariate gaussian distribution

From this we can estimate the inverse covariance matrix as given above. The estimate of the inverse covariance matrix is $hat{Sigma}^{-1}$.

Now the derive the dual of this problem

I didn't get how the dual was derived. Can anyone please explain? I would really appreciate it