I am trying to calculate the hessian of the log of the matrix-t distribution. I know that the log of the matrix-t distribution can be written: $$\log T_{N\times P}(X| \nu, M, \Sigma, \Omega) \propto -\frac{\nu + n+ p-1}{2}\log|I_N + \Sigma^{-1}(X-M)\Omega^{-1}(X-M)^T|$$ where $\nu$ is a scalar, $M$ is an $N\times P$ matrix, $\Sigma$ is a $N\times N$ covariance matrix and $\Omega$ is an $P\times P$ covariance matrix. I also believe that I can calculate the gradient and that it is given by:
$$\frac{d\log T_{N\times P}(X| \nu, M, \Sigma, \Omega)}{dX} \propto -c(\Sigma^{-1}+\Sigma^{-1}(X-M)\Omega^{-1}(X-M)^T\Sigma^{-1})^{-1}(X-M)\Omega^{-1}M^T$$ where I have let $c=\nu + n+ p-1$.
My issue, I don't know how to massage this into a form to calculate the Hessian. In particular, I am interested in the hessian in the form $$\frac{d\log T_{N\times P}(X| \nu, M, \Sigma, \Omega)}{dvec(X)dvec(X)'}$$ although I am flexible.