Could someone please verify that I got this problem correct or not:
Let $f: M(n,\mathbb{R}) \to M(n,\mathbb{R})$ defined by $f(A)=A^T+A.$ Calculate $Df_A$
Mt attempt:
Consider differential map $f_{*,A}: T_A M(n,\mathbb{R}) \to T_{f(A)} M(n,\mathbb{R})$. Note that $T_{A}M(n,\mathbb{R}) \cong \mathbb{R^{n^2}}$. Now (using a result) for any matrix $X\in \mathbb{\mathbb{R^{n^2}}}$, there is curve $c(t)$ in $M(n,\mathbb{R})$ with $c(0)=A$ and $c'(0)=X$, note the following:
$f_{*,A}(X)= \frac{d}{dt}f(c(t))|_{t=0}=\frac{d}{dt}(c(t)^T + c(t))|_{t=0}=(c'(t)^T + c'(t) )|_{t=0}=c'(0)^T+c'(0)=X^T+X$.
Hence $Df_A=X^T+X$. Does it make sense? or did I miss something.
Thanks for any insights.