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Not a complete answer, but I hope it steers you towards it. I'll show how it's possible to keep the matrices after the first derivative: $$\frac{\partial}{\partial\rho}\ln f_{\rho}(\boldsymbol x)= -\frac12\frac{\partial}{\partial\rho}\ln|\Sigma| - \frac12 \frac{\partial}{\partial\rho}\left(\boldsymbol x'\Sigma^{-1}\boldsymbol x\right) = -\frac12\text{tr}((\... 3 I find the literature in MLE a bit fuzzy with nomenclature here, so I might have some stuff off, and I will try to stick to the nomenclature you introduced. We have the observed Fisher information:$$\left[\mathcal {J}(\theta)\right]_{ij} = -\left(\frac{\partial^2 \log f}{\partial \theta_i \partial \theta_j}\right)$$And the empirical Fisher information:$$\...