Let $A$ be a $n \times n$ real matrix and $f:\Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}$ such that $ f(x,y)= \langle Ax,y \rangle$, where $ \langle x,y \rangle$ denotes the inner product of $x$ and $y$. Let $Df(x,y)$ denote the derivative of $f$ at $(x,y)$ which is a linear transformation from $\Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}$.

My question is how to find $Df(x,y)?$ I tried to use jacobian but I cannot.Please give me a hint to solve


$F$ is bilinear, so $DF_{(x,y)}(u,v)=\langle A(x),v\rangle+\langle A(u),y\rangle$.

A way to see this is to look at the partial derivative relatively to $x$ and $y$ of $F$.

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  • $\begingroup$ Thanks for the reply, but I cannot understand. Can you explain a bit more please. THank you $\endgroup$ – Sam Christopher Jun 16 '19 at 3:52
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    $\begingroup$ @SamChristopher Refer to Chapter 3, in particular Lemma 8.3 and Theorem 8.4 of Loomis and Sternberg's book (available online legally for free) for a statement and proof of how to compute derivatives of bilinear mappings. math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf $\endgroup$ – peek-a-boo Jun 16 '19 at 4:21

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