Let $F(m)=m^T m$ with $m$ a $n\times n$ matrix. I came across the statement $D_I F(m)=m^t+m$, where $D_iF$ means the derivative of $F$ at the identity matrix.
I cannot understand how this emerges. I can imagine taking the directional derivative of $F$ at $I$ by $\lim_{\lambda\rightarrow0} \frac{F(I+\lambda m)-F(I)}{\lambda}=\lim_{\lambda\rightarrow0} \frac{\lambda (m^T+m)+\lambda^2m^T m}{\lambda}=m^T+m.$
Is that the meaning of $D_I F(m)$? Is there a way to define a "general" derivative of a map from the space of matrices to the space of matrices?