What is the matrix $\left[ DS(A) \right]$, which gives $\left[ DS(A) \right] H=AH+HA$? In Hubbard's multivariable calculus book $DS(A):H \mapsto AH+HA$ is 
introduced as a linear transformation where $A$ is an $n \times n$ matrix, 
$S(A)=A^2$, and $D$ is the notation for derivative. 
It is used to prove that $S$ has a derivative if $\left[ DS(A) \right] H=AH+HA$ using the following limit:
$$\lim_{H \to 0}\frac{1}{||H||}||(S(A+H)-S(A))-\left[ DS(A) \right] H||=0$$
This limit really gives $0$, so S has a derivative. But the actual derivative matrix $\left[ DS(A) \right]$ is not given in the book and I can't see what kind of a matrix gives $AH+HA$ when it is multiplied by $H$. What is this matrix?

Using vectorized form $H_v:=\mbox{vec}(H)$ and the formula
    $\mbox{vec}(AB)=(I\otimes A)\mbox{vec}(B) =(B^{T}\otimes I)\mbox{vec}(A)$ we can write both terms with the $H_v$ on the right
$$
\mbox{vec}(AH+HA)=(I\otimes A)H_v + (A^T\otimes I)H_v,
$$
and find the $n^2\times n^2$ matrix to be  $I\otimes A + A^T\otimes I.$
