Derivative with respect to the variance of Multivariate Gaussian

Asked Jan 20 '20 at 11:51

Active Jan 20 '20 at 11:58

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I am working on a multivariate Gaussian distribution with mean $\mu \in \mathbb{R}^d$ and the variance-covariance $\Sigma \in S^{d\times d}$ where $S$ denotes the set of symmetric matrices.

When I am trying to maximize the log-likelihood of some observations (with respect to $\Sigma)$, I am basically trying to take the first-order conditions, and I need to derive the following:

$$\nabla_{\Sigma} \left( \mu^\top \Sigma^{-1}\mu \right)$$

This derivative is too complicated for me since $\Sigma$ is a matrix.

How can I solve these?

edited Jan 20 '20 at 11:58

Rodrigo de Azevedo

asked Jan 20 '20 at 11:51

independentvariable

@RodrigodeAzevedo the distribution is multivariate -- I mean variance-covariance matrix – independentvariable Jan 20 '20 at 11:57
Related: https://math.stackexchange.com/q/3330806/339790 – Rodrigo de Azevedo Jan 20 '20 at 11:59
@RodrigodeAzevedo sorry for duplicate. Thank you! – independentvariable Jan 20 '20 at 12:03
1

No need to apologize. You're most welcome. – Rodrigo de Azevedo Jan 20 '20 at 12:05

Derivative with respect to the variance of Multivariate Gaussian

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