I saw an equation like this...
$$ \frac{\partial}{\partial \mathbf{y}}\mathbf{y}^T\mathbf{\Sigma^{-1}}\mathbf{y} = (\mathbf{\Sigma^{-1}} + \mathbf{\Sigma^{-T}})\mathbf{y} $$
I am wondering about the reasoning behind ending up with $\Sigma^{-1} + \Sigma^{-T}$ instead of just $2\Sigma$. Thanks
Let $(\sigma_{ij})$ denote the entries of the matrix ${\bf{\Sigma}^{-1}}$. Then, we have: $${\bf{y}^{T}\Sigma^{-1}y} = \sum_{i,j}y_{i}\sigma_{ij}y_{j}$$ Now, for a fixed $k$, we have: $$\frac{\partial}{\partial y_{k}}\sum_{i,j}y_{i}\sigma_{ij}y_{j} = \sum_{j}\sigma_{kj}y_{j} + \sum_{j}\sigma_{jk}y_{k} = \sum_{j}(\sigma_{kj}+\sigma_{jk})y_{j}.$$ Thus, you'll have $2{\bf{\Sigma}^{-1}}$ instead of ${\bf{\Sigma^{-1}+\Sigma^{-T}}}$ if your matrix ${\bf{\Sigma}}^{-1}$ is symmetric.