Let: $\eta = \begin{bmatrix} \eta_1 \\ \eta_2 \\ \eta_3 \\ \eta_4 \end{bmatrix}$ and its derivative in respect to time: $\dot{\eta} = \frac{d\eta}{dt} = \begin{bmatrix} \dot{\eta_1} \\ \dot{\eta_2} \\ \dot{\eta_3} \\ \dot{\eta_4} \end{bmatrix}$.
Let $C$ be a $4 \times 4$ symmetric matrix whose elements are all functions of $\eta_1$, $\eta_2$, $\eta_3$ and $\eta_4$.
I have the following system of 4 equations (one for each element):
$\begin{cases} a_1 = \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_1}\right) \dot{\eta} \\ a_2 = \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_2}\right) \dot{\eta} \\ a_3 = \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_3}\right) \dot{\eta} \\ a_4 = \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_4}\right) \dot{\eta} \\ \end{cases}$
I need to rewrite this system in Matrix form, so I defined $A = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix} $ and rewrote the equation as:
$A = \begin{bmatrix} \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_1}\right) \dot{\eta} \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_2}\right) \dot{\eta} \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_3}\right) \dot{\eta} \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_4}\right) \dot{\eta} \\ \end{bmatrix}$
And managed to show it is equivalent to:
$A = \begin{bmatrix} \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_1}\right) \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_2}\right) \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_3}\right) \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_4}\right) \\ \end{bmatrix} \dot{\eta}$
It is possible to rewrite the expression $\begin{bmatrix} \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_1}\right) \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_2}\right) \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_3}\right) \\ \dot{\eta}^T \left(\cfrac{\partial C}{\partial \eta_4}\right) \\ \end{bmatrix}$ in a more simplified/compact way, defining maybe some kind of tensor operation or matrix derivative operator?
Thanks a lot in advance!
$\def\v{\operatorname{vec}}\def\p#1#2{\frac{\partial #1}{\partial #2}}$Define the scalar $$\eqalign{ \phi &= \dot\eta^T C\dot\eta \\ &= (\dot\eta\otimes\dot\eta)^T \v(C) \\ &= v^T c \\&= c^Tv \\ }$$ where $\otimes$ denotes the Kronecker product and the vec operation stacks the columns of $C$ to create one long column vector $c$.
The vector in question is the gradient of this scalar, i.e. $$\eqalign{ A &= \p{\phi}{\eta} &= \left(\p{c}{\eta}\right)^Tv \\ A_k &= \p{\phi}{\eta_k} &= \left(\p{c}{\eta_k}\right)^Tv \,\;=\; \dot\eta^T\left(\p{C}{\eta_k}\right)\dot\eta \\\\ }$$ NB: This assumes that $\{\eta,\dot\eta\}$ are independent variables, such as position and velocity.