1
$\begingroup$

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

$\endgroup$
3
$\begingroup$

$F$ 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 $F$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.