I would like to work out the following formulation. Consider a function $f($S$)$, where S is a matrix. If I want to compute the second order derivative of the function I have:
$$ G_{i jkl}=\frac{\partial f(S)}{\partial S_{i j} \partial S_{k l}}$$
and hence I can write the differential df(S) as:
$$df(S)= \sum_{ij} \sum_{kl} G_{ijkl} \partial S_{ij} \partial S_{kl}= \sum_{ij} \partial S_{i j} \langle G_{ij}, \partial S \rangle_{F}$$ How can I continue to write the above equation in a "quadratic" form similar to $xAx$ for vector-matrix-vector but now with matrix-tensor-matrix?
Thank you