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Suppose I have a function $\phi(M):=M^n$ where $M\in M_n(R)$

Does it make any sense to talk about the derivative of this function? What would it mean if it does make sense? I am also wondering what the derivative (if it exists and is meaningful) would be. (Perhaps $nM^{n-1}$?)

Thanks.

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1  
Look up Gâteaux and Fréchet derivatives. –  J. M. Nov 22 '11 at 0:55
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I'd just like to say that this is a great question - keep on thinking about how ideas you've learned in different mathematical contexts might interact! –  Zev Chonoles Nov 22 '11 at 1:26

2 Answers 2

Yes, it makes sense to talk about derivatives of functions $f:V\to W$, where $V$ and $W$ are a normed vector space, for example.

In this case, we say that $f$ is differentiable at $x\in V$ if there is a linear mapping $Df(x):V\to V$ such that

$$ \lim_{h\to 0}\frac{\|f(x+h)-f(x)-Df(x)h\|}{\|h\|}=0 $$

In your particular case, you could compute the derivative of $M\mapsto M^n$ by using a $n$-linear application given by $$ \varphi(A_1,\ldots,A_n)=A_1\cdot\ldots\cdot A_n. $$

You can show for any $X=(X_1,\ldots,X_n)\in M_n(R)\times\ldots\times M_n(R)$ ($n$-times) that

$$ D\varphi(X)(H_1,\ldots,H_n)=\sum_{i=1}^n \varphi(X_1,\ldots,X_{i-1},H_i,X_{i+1},\ldots,X_n). $$

So the formula you want should be interpreted as the derivative of $\varphi$ at $X=(M,\ldots,M)$ applied at $(Id,\ldots,Id)$. Which is equivalent to what we do when we write down this formula for real and complex functions.

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Yes, I think Frechet wrote about this originally in his paper: "Not just fresh, but Frechet" –  gary Nov 26 '11 at 6:24

Layman version:

Start with $n=1$: $f(M) = M$. Then $Df(M) = DM = I$.

For $n=2$: $f(M) = M*M$. Apply product rule: $$Df(M) = (DM) * M + M * (DM) = I*M + M*I = 2M$$

Induction on general $n\geq2$, assuming $D(M^{n-1}) = (n-1)*M^{n-2}$: $f(M) = M*M^{n-1}$. Product rule: $$Df(M) = (DM) * M^{n-1} + M * D(M^{n-1}).$$ Use induction: $$D(M^{n-1}) = (n-1)M^{n-2}.$$ Then $$Df(M) = I * M^{n-1} + M*(n-1)*M^{n-2} = n*M^{n-1}.$$

QED.

Of course, I'm assuming that the product rule holds.

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You can use LaTeX code to write math expressions on this site; here is a guide on how to do so. –  Zev Chonoles Nov 22 '11 at 1:30
    
@ZevChonoles Great, thanks a lot! –  Sjoerd Nov 22 '11 at 1:32
    
No problem! I should also mention that if you see a piece of LaTeX you want to know the code for on the site, you can right click on it and choose "Show Source" - this is a good way of picking up how to do things. –  Zev Chonoles Nov 22 '11 at 1:34

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