TITLE: Find differential for $A \mapsto A^2$
QUESTION [1 upvotes]: Let $f: \mathrm{Mat}_n(\mathbb{R}) \to \mathrm{Mat}_n(\mathbb{R})$ by rule $A \mapsto A^2$. How to find $\mathrm{d}f$?

REPLY [1 votes]: Hint: if $A\in \text{Mat}_n(\mathbb R)$ then for any $n\times n$ matrix $H$, you get $$f(A+H)- f(A) = AH+HA + H^2.$$

REPLY [1 votes]: Hint. You could consider
$$
(A+H)^2-A^2=(A^2+A\cdot H+H\cdot A+H^2)-A^2=A\cdot H+H\cdot A+o(||H||).
$$
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