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I'm trying to calculate a MLE and was wondering if anyone had a reference or could help me with derivatives involving matrices:

$$ \frac{\partial}{\partial \sigma^2} \sum\limits_{k = 1}^{K} (Y_k - \mu)^T\frac{1}{\sigma^2}(I_n - \frac{\tau^2}{\sigma^2 + n\tau^2}1_n)(Y_k - \mu) $$ or $$ \frac{\partial}{\partial \sigma^2} \sum\limits_{k = 1}^{K} (Y_k - \mu)^T\frac{1}{\sigma^2}(I_n - \frac{\tau^2}{\lambda}1_n)(Y_k - \mu) $$ where $$ \lambda = \sigma^2 + n\tau^2$$

Basically this is the last term in a log-likelihood of a multivariate normal, $$MVN(\mu, \sigma^2 I_n + \tau^2 1_n)$$

EDIT: $$Y,\mu, I_n, 1_n \text{ are matrices.}$$ $$\sigma^2, \tau^2\text{ are scalars.}$$

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    $\begingroup$ I don't see any matrix variables anywhere in this post: could you please explain what your notation means? $\endgroup$
    – whuber
    Commented Feb 19, 2016 at 22:01
  • $\begingroup$ Since you have revealed that $\sigma^2$ is a scalar, then there are no matrix derivatives anywhere in this question. In fact, once you simplify the sums you will discover that there aren't any matrices involved in them anyway. Ordinary (one-variable!) Calculus is all you need to know. $\endgroup$
    – whuber
    Commented Feb 19, 2016 at 22:23

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I and many of my friends have struggled with this exact problem. There's indeed a dearth of good resources and practice problems on matrix calculus! Continuum mechanics books tend to have a great deal of info on this topic. I suggest for example "A First Course in Continuum Mechanics" by Gonzalez and Stuart. Also, "The Matrix Cookbook" (https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) is a great reference for formulae from matrix calculus

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