How to find derivative of Multivariable function

Question

I am reading from multivariable calculus notes and need help with following:

Let $f : M(n,\mathbb{R}) \to M (n,\mathbb{R})$ and let $f(A)= AA^t$. Then find derivative of f, denoted by df .

So, Derivative of f (df) if exists, will satisfy $\lim H\to 0 \frac{||f(A+H) -f(A) -df(H)||}{||H||} =0$.

In these kind of questions how should I find what df(H) should be so that it has to be used to check the definition of derivative?

Should I go by guess work? What if there are more than 1 such df?

What you call the derivative is actually the differential of $f$. — Bernard, Nov 01 '21 at 19:56
https://math.stackexchange.com/questions/3993446/derivative-of-linear-map — , Nov 05 '21 at 08:14

score 0 · Accepted Answer · answered Nov 01 '21 at 19:59

0

We can try calculating $f(A+H)-f(A)$ and see what is left

$f(A+H)-f(A)= (A+H)(A+H)^t -AA^t= AA^t +AH^t + HA^t +HH^t- AA^t\\ =AH^t + HA^t +HH^t$.

The term $HH^t$ has norm $\|HH^t\|=\|H\|^2$ and so it will go to zero even when divided by $\|H\|$. Thus our candidate is

$df_{A}(H)=AH^t+HA^t$.

Now you can try and prove the limit is $0$ explicitly.

answered Nov 01 '21 at 19:59

podiki

why can't there be any other candidate? – Nov 02 '21 at 07:40
Can you please reply to my comment? – Nov 05 '21 at 08:34
See this post https://math.stackexchange.com/questions/611254/proof-that-the-derivative-is-unique/611424 – podiki Nov 05 '21 at 13:26

1 Answers1