I am stuck at this part in my notes and I just cant proceed any further from this point because I cant seem to understand this, and it goes:
If we have some inclusion map $i:\text{Sym}(n, \mathbb{R})\rightarrow M_{n\times n}\mathbb{R}$, it says that: $$\frac{\partial}{\partial x_{ij}}(i(F(\mathbb{R^{\frac{n(n+1)}{2}}})))=\frac{1}{2}(E_{ij}+E_{ji})$$
Where $F:\mathbb{R^{\frac{n(n+1)}{2}}} \rightarrow \text{Sym}(n, \mathbb{R})$, $x_{ij} \in R^{\frac{n(n+1)}{2}}$, such that $1\le i \le j \le n$. I have no idea why this is the case, also:
Define $\Phi: M_{n\times n} \rightarrow \text{Sym}(n, \mathbb{R})$, that is $A$ goes to $A^TA$:
$$\frac{\partial}{\partial x_{ij}}(A^TA)=(A^TE_{ij})^T+A^T E_{ij}$$ where $E_{ij}$ are $n \times n$ matrices with $1$ in the $(i,j)$ th entry, and $0$ elsewhere.
I have been scratching my head the past 3 hours trying to understand why this is the case. i would appreciate some help.