Prefacing apology for likely having unclear notation in the question and possible unclear concepts, because I'm not a mathematician.

The Fisher Information Matrix (FIM) for a multivariate normal distribution with $\mu$ and $\Sigma$ parameters is: 

\begin{equation}
\tag{1}
I_{\mu,\Sigma} = 
\left[
\begin{array}{cc}
\Sigma^{-1} & 0 \\
0 & \frac{1}{2}\Sigma^{-1}\Sigma^{-1}\\
\end{array}
\right]
\end{equation}

According to wikipedia's section on FIM [reparameterization][1], if a distribution $P({z};\bf{\theta})$ has a parameter vector of $\bf{\theta}$ and a FIM of $\bf{I}_{\bf{\theta}}$, then given continuous functions $\bf{\theta}(\bf{\eta})$ the FIM under reparameterization to the $\bf{\eta}$ parameter vector is given as:

\begin{equation}
\tag{2}
\bf{I}_{\bf{\eta}} = \bf{J}^{T} I_{\bf{\eta}} (\bf{\theta} (\bf{\eta})) \bf{J}
\end{equation}

Where $\bf{J}$ is the Jacobian matrix of $\bf{\theta}(\bf{\eta})$. This represents a change of coordinates essentially on the distribution's manifold.

Later on in the article, the [multivariate normal distribution][2] is discussed and the $(m,n)$ element of it's FIM is given by:

\begin{equation}
\tag{3}
I_{m,n} = \frac{\partial \mu}{\partial \theta_{m}}^{T} \Sigma^{-1} \frac{\partial \mu}{\partial \theta_{n}} + \frac{1}{2} \text{tr}\left(\frac{\partial \Sigma}{\partial \theta_{m}} \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_{n}} \Sigma^{-1} \right)
\end{equation}

where $\text{tr}(.)$ is the matrix trace, and $\mu$ and $\Sigma$ can be functions of $\theta$.

I've found a derivation of equation (3) here ([Result R37 on page 360][3]) (which seems to be publicly available from the author), but it doesn't involve the Jacobian as far as I can tell.


Although the Jacobian method of getting the reparametrized FIM is specified on wikipedia to only be for *vector* parameters (and $\Sigma$ is not a vector), I was wondering whether there's any extension/derivation involving it to go from the form in equation (1) to the form in equation (3) where $\mu$ and $\Sigma$ are functions of $\theta$? It would be great to have a general method that could work for any reparameterization step, which I think ought to be possible since it is a change of coordinates essentially.




  [1]: https://en.wikipedia.org/wiki/Fisher_information#Reparameterization
  [2]: https://en.wikipedia.org/wiki/Fisher_information#Multivariate_normal_distribution
  [3]: https://user.it.uu.se/~ps/SAS-new.pdf