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, 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 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) (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 it can still be applied to go from equation (1) to 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.
