Sorry for the basic query. I would like to understand the difference between the differentiation of an expression with respect to a matrix and with respect to a symmetric matrix.
Suppose that I have the following symmetric matrix
$M = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$,
and I would like to differentiate $f = a \, c - b^2$ with respect to this symmetric matrix.
According to the referecne, which does not provide detailed explanation, the solution is $\begin{bmatrix} c & -b \\ -b & a \end{bmatrix}$, but how?
Thanks in advance.