How to calculate derivative by element from matrices

Question

I have a matrix $\Sigma$ given by \begin{equation*} \Sigma = \begin{bmatrix} \phi_{11} &\phi_{12}\\ \phi_{12} & \phi_{22} \end{bmatrix} \end{equation*}

I'd like to calculate $\frac{\partial y^T\Sigma y}{\partial \phi_{11}}$,where $y$ is a vector.

Answer 1

You have: $$\partial y^T\Sigma y=\phi_{11} y_1^2+(\phi_{12}+\phi_{21}) y_1y_2+\phi_{22}y_2^2$$ so: $$\frac{\partial y^T\Sigma y}{\partial \phi_{11}}=y_1^2=y^T A y$$ where: $$A=\begin{bmatrix}1&0\\0&0 \end{bmatrix}$$